APEX Calculus PDF Download – Gregory Hartman (Version 4.0)

BS Math and BS Engineering students can download APEX Calculus by Gregory Hartman (Virginia Military Institute) for free — the complete three-volume textbook, not a preview. It covers the standard three-semester Calculus I–III sequence: limits and derivatives, integration and its applications, sequences and series, and on through parametric equations, vectors, multivariable functions and vector analysis — the same syllabus ground most BS programmes require across their calculus courses.

What sets APEX Calculus apart is how it frames the material: real-world setups drive most sections — a falling amusement-park ride motivates the derivative, a car submerged in water motivates fluid-force integrals — and 3,485 end-of-section exercises open with short conceptual questions before the computational ones. For BS Engineering students especially, the applications-first approach in Chapters 4 and 7 (related rates, optimization, work, fluid forces) connects directly to problems engineering courses actually pose, and Version 4.0’s 3D-interactive PDF graphics make Calculus III’s surfaces and solids easier to visualize than a static picture allows.

Book Overview

Chapter List

Chapter 1: Limits

Volume 1: Calculus I · Difficulty: Medium · Key topics: approximating limits graphically and numerically, the formal epsilon-delta definition, limit laws, one-sided limits, continuity, the Intermediate Value Theorem, limits involving infinity, horizontal and vertical asymptotes

The book opens by approximating limits graphically and numerically — using the example sin(x)/x near x=0, where direct substitution gives the indeterminate form 0/0 — before giving the formal epsilon-delta definition and the algebraic limit laws that make limits computable without a table or graph. One-sided limits and the three concrete ways a limit can fail to exist follow (different left/right values, unbounded growth, oscillation), then continuity and the Intermediate Value Theorem, and finally limits at infinity, horizontal asymptotes, and infinite limits at a vertical asymptote.

Key Points:

• A limit describes the value a function approaches near a point, never the function’s actual value there — f(c) can differ from lim(x→c) f(x) or not exist at all.
• Epsilon comes first: a limit proof fixes an output tolerance ε, then finds an input tolerance δ that guarantees it — not the other way around.
• A limit fails to exist in exactly three ways: the left- and right-hand limits disagree, the function grows without bound, or it oscillates without settling.
• A function is continuous at c when lim(x→c) f(x) exists and equals f(c) — both conditions are required, and neither alone is enough.
• The Intermediate Value Theorem guarantees a value exists between f(a) and f(b) on a continuous interval; it proves existence without showing how to find it.
• lim(x→∞) f(x) = L means y = L is a horizontal asymptote; a vertical asymptote comes from a limit that grows without bound as x approaches a specific value.

Memory Tip: Read an epsilon-delta proof as a negotiation that starts with the reader: “you hand me how close you want the output (ε), and I hand back how close the input needs to be (δ).” Getting the order backwards is the single most common way to misstate the definition.

Common Mistake: Assuming a limit exists just because the function is defined at that point, or assuming it does not exist just because the function is not defined there — the book’s own opening example, sin(x)/x at x=0, is undefined at 0 yet still has a limit of 1 there.

Important Questions:

• Which is given first in establishing a limit, the x-tolerance or the y-tolerance? The y-tolerance (epsilon) is given first; the x-tolerance (delta) is then found in response to it.
• True or false: if f is continuous at c, then lim(x→c) f(x) exists? True — continuity at a point requires the limit to exist there (and to equal f(c)).

Chapter 2: Derivatives

Volume 1: Calculus I · Difficulty: Medium · Key topics: the derivative as a limit of a difference quotient, instantaneous vs. average rate of change, differentiability, basic differentiation rules, the Product and Quotient Rules, the Chain Rule, implicit differentiation, derivatives of inverse functions

Where Chapter 1 asked “where is a function going,” this chapter asks “how fast.” It defines the derivative as the limit of a difference quotient, motivated by a falling amusement-park ride whose exact velocity at a chosen instant cannot be read off an average speed. From there the chapter builds the mechanical toolkit needed to differentiate almost anything encountered in the rest of the book: the constant, power, sum and basic trig rules, then the Product and Quotient Rules, the Chain Rule for composed functions, implicit differentiation for curves not solved for y, and the rule for differentiating an inverse function without first finding a formula for it.

Key Points:

• The derivative at a point is lim(h→0) [f(x+h) - f(x)] / h — the limit of the average rate of change over a shrinking interval.
• A function differentiable at a point is automatically continuous there, but continuity does not guarantee differentiability — a sharp corner is continuous yet has no derivative.
• The Product Rule is (fg)′ = f′g + fg′; the Quotient Rule is (f/g)′ = (f′g - fg′)/g² — neither is the derivative of each factor multiplied or divided together.
• The Chain Rule differentiates a composition “outside-in, then multiply by the inside’s derivative”: d/dx[f(g(x))] = f′(g(x))·g′(x).
• Implicit differentiation applies d/dx to both sides of an equation in x and y together, treating y as a function of x and applying the Chain Rule to every y-term.

Practice Tip: Before reaching for the Chain Rule on a composed function, name the “outside” function and the “inside” function out loud first — most Chain Rule errors come from differentiating the outside function but forgetting to multiply by the inside function’s own derivative.

Common Mistake: Applying the Power Rule to an exponential expression like 2ᴏ, treating it as if the variable were the base instead of the exponent — the Power Rule only differentiates a fixed power of a variable base, never a fixed base raised to a variable power.

Important Questions:

• What does the definition of the derivative of a function at a point involve? Taking a limit — specifically the limit of the difference quotient as the interval shrinks to zero.
• Is it true that the average rate of change on [a,b] is the slope of the line through (a, f(a)) and (b, f(b))? True — that line is the secant line, and its slope is exactly the average rate of change over that interval.

Chapter 3: The Graphical Behavior of Functions

Volume 1: Calculus I · Difficulty: Medium · Key topics: absolute and relative extrema, the Extreme Value Theorem, the Mean Value Theorem, Rolle’s Theorem, increasing and decreasing functions, concavity, the second derivative, curve sketching

Having built differentiation rules in Chapter 2, this chapter puts them to work reading a function’s own shape off its derivatives. It opens with absolute and relative extrema and the Extreme Value Theorem (a continuous function on a closed interval must attain a max and a min), then the Mean Value Theorem — guaranteeing some point where the instantaneous rate matches the average rate — and its special case, Rolle’s Theorem. The first derivative’s sign gives increasing/decreasing behavior and locates extrema; the second derivative’s sign gives concavity and inflection points. The chapter closes by combining all of this into a full curve-sketching method.

Key Points:

• An absolute extremum is the largest/smallest value on the whole domain being considered; a relative extremum is only the largest/smallest in some neighborhood around it.
• The Extreme Value Theorem guarantees a continuous function on a closed interval [a,b] attains both an absolute maximum and minimum somewhere on that interval.
• The Mean Value Theorem guarantees a point c in (a,b) where f′(c) equals the average rate of change over [a,b] — Rolle’s Theorem is the special case where f(a)=f(b) and the guaranteed point has f′(c)=0.
• A relative extremum can only occur at a critical point — where f′(x)=0 or f′(x) does not exist — though not every critical point is an extremum.
• f″(x) > 0 means the graph is concave up (curving like a cup); f″(x) < 0 means concave down; a sign change in f″ marks an inflection point.

Memory Tip: Keep the derivative order straight: the first derivative tells you which way the graph is heading (increasing/decreasing); the second derivative tells you how it’s bending (concave up/down). Sketch a simple cubic and check both signs against the picture until this is automatic.

Common Mistake: Assuming every point where f′(x)=0 is automatically a relative extremum — a critical point can also be neither, as with f(x)=x³ at x=0, where the derivative is zero but the graph merely flattens without turning.

Important Questions:

• What does the Mean Value Theorem state, in your own words? That on a closed interval where a function is continuous and differentiable, there is some point where the instantaneous rate of change equals the average rate of change over the whole interval.
• What is the difference between absolute and relative maxima? An absolute maximum is the largest value the function attains over its entire domain (or the interval under consideration); a relative maximum is only the largest value in some neighborhood around that point.

Chapter 4: Applications of the Derivative

Volume 1: Calculus I · Difficulty: Hard · Key topics: Newton’s Method, related rates, optimization word problems, differentials and linear approximation

This chapter turns the derivative toolkit into a problem-solving toolkit. Newton’s Method uses tangent lines to approximate roots of equations that cannot be solved algebraically, such as cos x = x. Related rates connects two quantities through an equation and uses the derivative to find how fast one changes given the rate of the other — the book’s own example starts from the plain geometry of a circle’s circumference and radius. Optimization applies extreme-value techniques to real word problems (maximizing area, minimizing cost). Differentials close the chapter, using the tangent line to approximate f(x+Δx) without evaluating f directly.

Key Points:

• Newton’s Method iterates x(n+1) = x(n) - f(x(n))/f′(x(n)), using the tangent line at each guess to land closer to a root of f(x)=0.
• Newton’s Method gives an approximate solution, not an exact one — and does not guarantee that a fixed number of iterations reaches a fixed number of correct decimal places.
• A related-rates problem always starts from an equation linking two or more quantities, then differentiates both sides with respect to time before substituting known values.
• Optimization word problems reduce to finding an absolute extremum of a function built from the problem’s own constraint equation, usually on a closed, bounded interval.
• The differential dy = f′(x)·dx approximates the actual change Δy using the tangent line, and the approximation gets worse the further Δx is from zero.

Practice Tip: On a related-rates problem, write down every given rate and every rate being asked for before touching the equation — most errors come from differentiating first and only then realizing a needed quantity was never assigned a variable.

Common Mistake: Substituting the known numeric values into the relating equation before differentiating it, instead of after — differentiating a number instead of a variable silently drops it from the rate equation entirely.

Important Questions:

• True or false: Newton’s Method produces an exact solution to f(x) = 0? False — it produces an approximate solution, converging closer to a root with each iteration but never guaranteed to land on it exactly.
• True or false: at least d+1 iterations of Newton’s Method must be used to get a solution accurate to d decimal places? False — there is no fixed relationship between the number of iterations and the number of correct decimal places reached; convergence speed depends on the function and the starting guess.

Chapter 5: Integration

Volume 1 & 2: Calculus I / II — Volume 1 covers this chapter in full · Difficulty: Medium · Key topics: antiderivatives, indefinite integration, the definite integral, Riemann Sums, the Fundamental Theorem of Calculus, numerical integration (Trapezoidal Rule, Simpson’s Rule)

The book now reverses direction: given f(x), find F(x) such that F′(x) = f(x). Antiderivatives and indefinite integration open the chapter, framed explicitly as solving the simplest possible differential equation. The definite integral is then built from Riemann Sums — approximating a signed area by rectangles and taking the limit as the rectangles narrow — before the Fundamental Theorem of Calculus connects the two halves of the chapter: definite integrals can be evaluated with antiderivatives instead of limits of sums. The chapter closes with numerical integration (Trapezoidal and Simpson’s Rules) for integrals with no elementary antiderivative.

Key Points:

• An antiderivative F of f satisfies F′(x) = f(x); because the derivative of a constant is 0, every antiderivative comes in a family F(x) + C.
• A Riemann Sum approximates a definite integral as ∑ f(c(i))Δx(i) over a partition of [a,b]; the definite integral is the limit of that sum as the partition’s width shrinks to 0.
• The Fundamental Theorem of Calculus (Part 1) states ∫(a to b) f(x)dx = F(b) - F(a) for any antiderivative F of f — the single most important shortcut in the whole book.
• The Fundamental Theorem (Part 2) states that d/dx[∫(a to x) f(t)dt] = f(x) — differentiation and integration are inverse processes.
• Simpson’s Rule fits parabolas instead of straight lines (Trapezoidal Rule) or flat tops (rectangles) to the partition, and is generally far more accurate for the same number of subdivisions.

Memory Tip: Say the Fundamental Theorem as a sentence, not a formula: “the definite integral of a rate of change over an interval equals the total change over that interval.” That framing is why the theorem connects area (Chapter 5’s subject) back to the derivative (Chapter 2’s subject).

Common Mistake: Forgetting the “+C” on an indefinite integral, or including one on a definite integral where it always cancels out — the book’s own exercises test exactly this distinction directly.

Important Questions:

• Is it more accurate to refer to “the” antiderivative of f(x) or “an” antiderivative of f(x)? “An” antiderivative — since any two antiderivatives of the same function differ only by a constant, there is no single unique antiderivative.
• How are definite and indefinite integrals related? The Fundamental Theorem of Calculus evaluates a definite integral by finding any indefinite integral (antiderivative) of the same function and computing the difference of its values at the two bounds.

Chapter 6: Techniques of Antidifferentiation

Volume 1 (Section 6.1 only) & Volume 2 (full chapter): Calculus I / II · Difficulty: Hard · Key topics: substitution, integration by parts, trigonometric integrals, trigonometric substitution, partial fraction decomposition, hyperbolic functions, L’Hôpital’s Rule, improper integration

Calculus I ends at Section 6.1 (Substitution, “undoing” the Chain Rule); Calculus II carries the full eight-section chapter. Integration by Parts follows, undoing the Product Rule for integrals like ∫x·eᴏdx. Trigonometric integrals and trigonometric substitution handle powers and products of sine/cosine and integrands containing √(a²-x²)-type expressions. Partial fraction decomposition rewrites a rational function as a sum of simpler fractions, motivated by an integral that has no obvious antiderivative until it is split. Hyperbolic functions and L’Hôpital’s Rule (for evaluating indeterminate-form limits) round out the toolkit before improper integration extends definite integrals to infinite or unbounded intervals.

Key Points:

• Substitution reverses the Chain Rule; Integration by Parts (∫u dv = uv - ∫v du) reverses the Product Rule.
• Partial Fraction Decomposition only applies to a proper rational function (numerator’s degree less than denominator’s) whose denominator factors into linear and irreducible quadratic pieces.
• L’Hôpital’s Rule applies only to the indeterminate forms 0/0 or ∞/∞ — applying it to a limit that is not already in one of those forms gives a wrong answer.
• An improper integral is defined as a limit: an infinite-interval integral becomes lim(b→∞) ∫(a to b), and one with an unbounded integrand becomes a one-sided limit approaching the bad point.
• Trigonometric substitution is chosen by matching the radical to a Pythagorean identity: √(a²-x²) → x=a sinθ, √(a²+x²) → x=a tanθ, √(x²-a²) → x=a secθ.

Practice Tip: Before picking a technique, ask what the integrand’s shape is — a product suggests Parts, a radical of the Pythagorean form suggests trig substitution, a rational function with a factorable denominator suggests partial fractions. Misidentifying the shape, not misapplying the technique itself, is where most time gets lost.

Common Mistake: Using polynomial long division before Partial Fraction Decomposition only when it’s “needed” by feel, instead of always checking first whether the numerator’s degree is ≥ the denominator’s — skipping this check when it is needed produces a decomposition that cannot be solved for its unknown constants.

Important Questions:

• Substitution “undoes” what derivative rule? The Chain Rule.
• True or false: it is sometimes necessary to use polynomial division before using Partial Fraction Decomposition? True — whenever the numerator’s degree is greater than or equal to the denominator’s degree, division must come first.

Chapter 7: Applications of Integration

Volume 2: Calculus II · Difficulty: Medium · Key topics: area between curves, volume by cross-sections, the Disk and Washer Methods, the Shell Method, arc length, surface area of revolution, work, fluid forces

Having built the definite integral in Chapter 5, this chapter applies the same “approximate with a Riemann Sum, then take the limit to get an exact integral” pattern to a run of genuinely different physical quantities. Area between two curves and volumes of revolution (Disk/Washer Method for cross-sections perpendicular to the axis, Shell Method for cylindrical shells parallel to it) come first, followed by arc length and surface area for curved paths and surfaces. The chapter closes with two physics applications the book frames memorably: work (the amount of force times distance needed to, say, empty a water tank) and fluid force (illustrated with the classic “can you open a submerged car door” problem).

Key Points:

• The Disk/Washer Method slices a solid of revolution perpendicular to the axis of rotation; the Shell Method uses cylindrical shells parallel to that axis — the axis and the shape of the region usually make one method far easier than the other.
• Arc length is ∫√(1+[f′(x)]²) dx — derived from the Pythagorean theorem applied to infinitesimally short line segments along the curve.
• Work for a constant force is W = F·d; for a variable force (like a spring or a draining tank), work becomes an integral of force over distance.
• Fluid force on a submerged surface integrates pressure (which increases with depth) times the width of the surface at each depth.
• Surface area of revolution integrates 2π·(radius)·(arc length element) around the axis of rotation.

Memory Tip: “Disk cuts across, shell wraps around” — say it before setting up a volume-of-revolution integral. Whichever method matches how the representative slice is oriented relative to the axis is almost always the easier setup.

Common Mistake: Setting up a volume-of-revolution integral with the wrong radius when the region does not touch the axis of rotation — the Washer Method’s outer and inner radii must both be measured from the axis, not from the edge of the region itself.

Important Questions:

• True or false: the area between curves is always positive? True — area between curves is computed as the integral of the absolute difference between the two functions, so it is defined to be non-negative regardless of which curve lies on top.
• True or false: calculus can be used to find the area of basic geometric shapes? True — the area-between-curves and disk/shell techniques recover the familiar formulas for triangles, circles, and other basic shapes as special cases.

Chapter 8: Sequences and Series

Volume 2: Calculus II · Difficulty: Hard · Key topics: sequences, infinite series, the Integral and Comparison Tests, the Ratio and Root Tests, alternating series and absolute convergence, power series, Taylor Polynomials, Taylor Series

The book itself flags this chapter as a genuine shift: almost no derivatives or integrals appear, but limits — especially limits involving infinity — are everywhere. A sequence is an ordered list of numbers; a series sums a sequence’s terms, and whether that sum settles on a finite value (“converges”) or not (“diverges”) is the chapter’s central question, tackled with a toolbox of six tests, each suited to a different shape of term. Power series extend this to series of functions, and the chapter’s payoff is Taylor Polynomials and Taylor Series — using a function’s derivatives at one point to build a polynomial (or infinite series) that approximates the function nearby, directly motivated by the question “if I know f(1), f′(1), f″(1)… what can I say about f(2)?”

Key Points:

• A series converges only if its sequence of terms converges to 0 — but a term sequence going to 0 does not by itself guarantee the series converges (the harmonic series is the classic counterexample).
• The Ratio Test is most effective on terms containing factorials and/or exponentials; it is not effective on terms that are purely algebraic or polynomial.
• The Alternating Series Test only requires the underlying sequence to be positive, decreasing, and heading to 0 — it says nothing about absolute convergence.
• Absolute convergence (the series of absolute values converges) is a stronger condition than convergence; a series that converges but not absolutely is conditionally convergent.
• A Taylor Polynomial of degree n at x=c uses f(c), f′(c), …, f⁽ⁿ⁹(c) to build the best polynomial approximation of that degree near c; the tangent line is exactly the degree-1 case.

Memory Tip: Keep a one-line “which test?” cheat sheet while working: factorial or exponential terms → Ratio Test; terms that compare cleanly to a known p-series or geometric series → Comparison Test; alternating sign → Alternating Series Test first, then check absolute convergence separately.

Common Mistake: Concluding a series converges just because its terms shrink toward 0 — the harmonic series ∑1/n has terms going to 0 yet diverges, which is exactly why a dedicated convergence test is always required, never just a glance at the terms.

Important Questions:

• The Ratio Test is not effective when the terms of a sequence only contain what kind of functions? Algebraic, or polynomial, functions.
• The domain of a sequence is which set of numbers? The natural numbers — a sequence is formally a function whose domain is the natural numbers (or a subset starting from some fixed integer).

Chapter 9: Curves in the Plane

Volume 3: Calculus III · Difficulty: Medium · Key topics: conic sections (degenerate and nondegenerate), parametric equations, calculus with parametric equations, polar coordinates, calculus with polar functions

Chapter 9 opens Calculus III by admitting that y=f(x) graphs are fundamentally limited — the vertical line test excludes shapes as basic as a full circle. Conic sections (circle, ellipse, parabola, hyperbola, plus their “degenerate” forms) are covered first through their geometric definitions. Parametric equations then describe curves by giving both x and y as functions of a third variable t, deliberately breaking the vertical-line restriction, with derivatives and arc length re-derived for this new form. Polar coordinates offer yet another way to plot points, using a distance and an angle instead of two perpendicular distances, with its own versions of slope, area and arc length.

Key Points:

• A nondegenerate conic (circle, ellipse, parabola, hyperbola) comes from a plane intersecting a double-napped cone away from its tip; a degenerate conic comes from a plane through the tip itself.
• A parametric curve (x(t), y(t)) can represent shapes like a full circle that a single function y=f(x) never can, since the vertical line test does not apply.
• The slope of a parametric curve is dy/dx = (dy/dt)/(dx/dt), not simply dy/dt or dx/dt alone.
• A polar point (r,θ) is not unique — the same point on the plane can be written with infinitely many different (r,θ) pairs, including negative r values.
• Of the conics, the hyperbola has the largest eccentricity (e > 1); a circle has the smallest (e = 0).

Memory Tip: Anchor eccentricity on a number line from 0: circle at e=0, ellipse between 0 and 1, parabola at exactly e=1, hyperbola above 1. The larger e gets, the more “open” and less “closed” the curve becomes.

Common Mistake: Assuming a polar point with a given r must have r > 0 — a negative r is valid and plots the point in the opposite direction from the given angle θ.

Important Questions:

• What is the difference between degenerate and nondegenerate conics? Nondegenerate conics are formed when a plane intersects a double-napped cone away from the cone’s tips (usually taken as the origin); degenerate conics are formed when the plane passes through the tip itself.
• What has the largest eccentricity: an ellipse or a hyperbola? A hyperbola.

Chapter 10: Vectors

Volume 3: Calculus III · Difficulty: Easy · Key topics: Cartesian coordinates in 3D space, quadric surfaces, introduction to vectors, the Dot Product, the Cross Product, lines and planes in space

This chapter moves the book’s mathematics out of the plane and into three-dimensional space, the foundation everything through Chapter 14 builds on. It starts by extending the familiar x–y plane to x–y–z space and introducing quadric surfaces (the 3D analogues of conic sections, including the saddle-shaped hyperbolic paraboloid). Vectors are then introduced as objects with both magnitude and direction — distinct from points — followed by the Dot Product (which measures the angle between two vectors and tests perpendicularity) and the Cross Product (which produces a new vector perpendicular to both inputs). The chapter closes using both products to describe lines and planes in space.

Key Points:

• In the plane, x=2 defines a line; in space, that same equation defines an entire plane — one fewer condition always describes one dimension more.
• A vector ⟨a,b,c⟩ and a point (a,b,c) use the same numbers but mean different things — a vector is a displacement, a point is a location.
• The Dot Product u·v = ‖u‖‖v‖cosθ is zero exactly when the two vectors are perpendicular.
• The Cross Product u×v produces a new vector perpendicular to both u and v, with magnitude equal to the area of the parallelogram they span.
• A plane in space is fully determined by one point on it plus one normal vector perpendicular to it.

Memory Tip: “Dot gives a number, cross gives a vector” — the Dot Product’s output is a scalar (useful for angles and perpendicularity), the Cross Product’s output is a new vector (useful for normals and areas). Confusing which operation returns which type of answer is the most common early error.

Common Mistake: Writing a vector and a point with the same coordinates and treating them interchangeably in a formula — a displacement vector from the origin looks identical to a point’s coordinates but behaves completely differently when added to another vector.

Important Questions:

• What is the difference between (1,2) and ⟨1,2⟩? (1,2) is a point; ⟨1,2⟩ is a vector describing a displacement of 1 unit in the x-direction and 2 units in the y-direction.
• Which quadric surface looks like a Pringles chip? A hyperbolic paraboloid.

Chapter 11: Vector-Valued Functions

Volume 3: Calculus III · Difficulty: Medium · Key topics: vector-valued functions, limits/derivatives/integrals of vector-valued functions, the calculus of motion (position, velocity, acceleration), unit tangent and normal vectors, arc length parameter, curvature

Building on Chapter 10’s vectors, this chapter defines functions whose output is a vector rather than a number — directly connected to parametric equations from Chapter 9, but now with the full machinery of calculus (limits, derivatives, integrals) applied component-by-component. The payoff is the calculus of motion: if r(t) is a position function, r′(t) is velocity and r″(t) is acceleration, with speed as the magnitude of velocity. The chapter then defines the unit tangent and unit normal vectors (which always point in the direction of travel and toward the “inside” of a turn, respectively) and closes with curvature — a number measuring exactly how sharply a curve bends at a point.

Key Points:

• A vector-valued function r(t) = ⟨f(t), g(t)⟩ (or with a third component for space) outputs a vector for every input t; its graph is traced by the tip of that vector as t varies.
• For a position function r(t): velocity is r′(t), acceleration is r″(t), and speed is the scalar ‖r′(t)‖.
• The unit tangent vector T(t) always has magnitude 1 by construction — it is r′(t) normalized to length 1, pointing purely in the direction of motion.
• The unit normal vector N(t) is always perpendicular to T(t), meaning N(t)·T(t) = 0.
• Curvature measures how quickly the unit tangent vector changes direction relative to arc length — a straight line has curvature 0; a tighter turn has higher curvature.

Memory Tip: Differentiate a vector-valued function exactly like a list — differentiate each component separately and keep the vector notation. There is no product or chain rule surprise here; it is component-wise calculus wearing vector notation.

Common Mistake: Confusing speed with velocity when reading a motion problem — velocity is the vector r′(t) (direction included), speed is the scalar ‖r′(t)‖ (direction discarded); a question asking “how fast” wants speed, not the velocity vector.

Important Questions:

• If T(t) is a unit tangent vector, what is ‖T(t)‖? 1 — that is exactly what “unit” vector means.
• If N(t) is a unit normal vector, what is N(t)·r′(t)? 0 — the unit normal vector is by definition perpendicular to the tangent direction, and r′(t) points in that same tangent direction.

Chapter 12: Functions of Several Variables

Volume 3: Calculus III · Difficulty: Hard · Key topics: multivariable functions, limits and continuity in 2D, partial derivatives, differentiability and the total differential, the multivariable Chain Rule, directional derivatives and the gradient, tangent planes, extreme values

Real quantities often depend on more than one input — the book’s own examples are wind chill (temperature and wind speed) and gas volume (pressure and temperature) — and this chapter builds calculus for functions z=f(x,y) of two (or more) variables. Partial derivatives hold all but one variable constant and differentiate normally; the total differential and a redefined notion of differentiability follow, since in several variables the existence of partial derivatives alone does not guarantee good local linear behavior. The multivariable Chain Rule, directional derivatives (rate of change in any direction, not just along an axis) and the gradient vector come next, then tangent planes, and finally extreme values — finding the largest/smallest output of a multivariable function, generalizing Chapter 3’s single-variable extrema.

Key Points:

• A function of two variables z=f(x,y) assigns each input pair (x,y) in its domain D a single output z; its graph is a surface in 3D space, not a curve.
• A partial derivative f(x) differentiates with respect to x while treating y as a constant coefficient — and vice versa for f(y).
• The gradient ∇f points in the direction of steepest increase of f at a point, and its magnitude gives the maximum possible rate of increase in any direction there.
• A relative extremum of z=f(x,y) can only occur at a critical point — where both partial derivatives are 0 or at least one fails to exist — exactly analogous to the single-variable case in Chapter 3.
• Level curves (used to draw a topographical map) are the 2D cross-sections formed by setting f(x,y) equal to a constant.

Memory Tip: Read ∂f/∂x as “freeze y, then differentiate as usual” — that single instruction is the entire idea behind a partial derivative, no matter how many variables the function has.

Common Mistake: Concluding a function is differentiable at a point just because both partial derivatives exist there — in several variables, existence of the partials is necessary but not sufficient; the full definition of differentiability (via the total differential) can still fail even when both partials exist.

Important Questions:

• The graph of a function of two variables is what kind of object? A surface (in three-dimensional space).
• Most people are familiar with the concept of level curves in the context of what kind of maps? Topographical maps.

Chapter 13: Multiple Integration

Volume 3: Calculus III · Difficulty: Hard · Key topics: iterated integrals, double integration and volume, double integrals in polar coordinates, center of mass, surface area, triple integration, cylindrical and spherical coordinates

Where Chapter 5’s definite integral gave “area under a curve,” this chapter’s double integral gives “volume under a surface” — and, just as single-variable integration had applications beyond area, multiple integration has applications beyond volume. Iterated integrals integrate one variable at a time, holding others constant exactly as in partial differentiation, with bounds that go “curve to curve, then point to point.” Double integration in polar coordinates handles circular regions more naturally than Cartesian bounds allow. Center of mass and surface area extend the technique to physical quantities, and the chapter closes with triple integration, including cylindrical and spherical coordinate systems for regions with rotational or spherical symmetry.

Key Points:

• An iterated integral evaluates one variable at a time; after integrating with respect to x, the “constant” of integration is really a function of y, since y was held fixed only for that step.
• Integrating an integral is called iterated integration, and its bounds follow the pattern “curve to curve, then point to point” for a 2D region.
• A double integral ∫∫(R) dA gives the plain area of region R; adding an integrand f(x,y) gives the volume under the surface z=f(x,y) above R.
• Polar double integrals require an extra factor of r in the area element (dA = r dr dθ), which is easy to forget when converting from Cartesian bounds.
• The strategy for triple-integral bounds extends the same pattern: “surface to surface, curve to curve, and point to point.”

Memory Tip: Say a triple integral’s bound-finding strategy out loud before setting one up: “surface to surface, curve to curve, point to point.” Getting the bounds in the wrong order is the most common way a correct-looking triple integral evaluates to the wrong number.

Common Mistake: Converting a double integral to polar coordinates and forgetting the extra factor of r in dA = r dr dθ — the integrand converts correctly but the area element itself is silently wrong without it.

Important Questions:

• When integrating f(x)(x,y) with respect to x, is the constant of integration C(x) or C(y)? What does this mean? C(y) — instead of being a plain constant, it is a function of y, which behaves like a constant only with respect to x since y was held fixed during that integration step.
• Give an informal interpretation of what a triple integral over a region D means. One natural reading is “sum up lots of little volumes over D.”

Chapter 14: Vector Analysis

Volume 3: Calculus III · Difficulty: Hard · Key topics: line integrals, vector fields, line integrals over vector fields, flow and flux, Green’s Theorem, the Divergence Theorem, parametrized surfaces and surface area, surface integrals, Stokes’ Theorem

Added in Version 4.0 (the book itself lists this as a headline change from Version 3.0), the final chapter connects vectors and integration in genuinely new ways — computing the work a magnetic field does moving an object along a path, or how much air flows through an oddly-shaped screen in space. Line integrals begin the chapter, both over scalar functions and over vector fields. Flow (movement along a curve) and flux (movement across a curve) are distinguished carefully, leading into Green’s Theorem, which relates a line integral around a closed curve to a double integral over the region it encloses. Parametrized surfaces, surface integrals, and finally the Divergence and Stokes’ Theorems — the 3D generalizations of Green’s Theorem — close both the chapter and the book.

Key Points:

• Flow measures the amount of a vector field F moving along a curve C; flux measures the amount moving across C — the same field integrated the same way can answer two genuinely different physical questions depending on which is asked.
• Circulation is the measure of flow around the entirety of a closed curve C.
• The ∮ notation (versus plain ∫) signals that the curve C is closed; the evaluation method itself is identical either way.
• Green’s Theorem converts a line integral around a closed curve into a double integral over the enclosed region — often turning a hard boundary computation into an easier area computation, or vice versa.
• Most line integrals are evaluated using the arc-length-parametrization Key Idea rather than directly, since the arc-length parameter itself is generally difficult to work with; parametrizing the curve with an easier parameter and converting is the standard approach.

Memory Tip: Keep flow and flux straight with direction words: flow is along the path (like a swimmer moving with a current), flux is across the path (like water passing through a net). The vector field and curve can be identical; only the question being asked changes.

Common Mistake: Treating ∮(C) f ds as requiring a different evaluation method from ∫(C) f ds just because of the closed-curve notation — the evaluation is identical; only the geometric meaning (that C is closed) differs.

Important Questions:

• What is circulation? The measure of flow around the entirety of a closed curve.
• How does evaluating a line integral written with ∮ differ from one written with plain ∫? It does not differ in evaluation — the ∮ notation only signals that the curve C being integrated over is closed; the computation method is the same either way.

Download APEX Calculus PDF (Free)

This book is free from its official source. Click below to open the official APEX Calculus site, where you can download the free PDF for whichever volume(s) you need — Calculus I, II or III.

How to Study This Book

Read the three volumes in their natural order — Calculus I (Chapters 1–6.1), Calculus II (Chapters 5–8), Calculus III (Chapters 9–14) — since each volume is written to assume everything before it. Within a volume, chapters build directly on each other: Chapter 2’s derivative rules are used constantly from Chapter 3 onward, and Chapter 5’s Fundamental Theorem is the hinge the entire second half of the book turns on.

If you are taking Calculus I this semester, download just that volume and read Chapters 1 through 4 in order, then Chapter 5 and Section 6.1. You do not need Calculus II’s copy of Chapters 5–6 — Calculus I already includes the same material through Section 6.1; only buy or download Calculus II when your course moves into Section 6.2 (Integration by Parts) and beyond.

If you are taking Calculus II, you can start directly at Chapter 5 if your Calculus I course used a different book — Chapters 5 and 6 are written to stand as a self-contained review-then-extend unit, and Chapter 7’s applications (area, volume, work, fluid force) all lean on Chapter 5’s Fundamental Theorem rather than anything specific to Chapters 1–4. Chapter 8 (Sequences and Series) is the chapter students most often underestimate; the book itself calls out that it uses almost no derivatives or integrals, so do not skip straight to the Ratio and Root Tests without first being comfortable with limits at infinity from Chapter 1.

If you are taking Calculus III (multivariable calculus), Chapters 9 through 11 (curves, vectors, vector-valued functions) build the geometric and vector language that Chapters 12 through 14 depend on completely — do not skip Chapter 10’s Dot and Cross Products, since Chapter 12’s gradient and Chapter 14’s flux both use them directly. Chapters 12 and 13 (multivariable functions, multiple integration) form one continuous unit; Chapter 14 (Vector Analysis, with Green’s, Stokes’ and the Divergence Theorems) is the hardest material in the entire book and was the last chapter added to the text, in Version 4.0.

What this book does not cover at all, confirmed directly against the Version 4.0 source: differential equations get no dedicated chapter (the book briefly uses the phrase “differential equation” only in passing, in Chapter 5, to motivate antiderivatives) and Lagrange multipliers do not appear in Chapter 12’s Extreme Values section. If your syllabus requires either topic, plan to supplement this book rather than expecting to find it here.

Do the exercises — there are 3,485 of them across the three volumes, and the book consistently opens each section’s exercise set with short conceptual questions (“in your own words, define…”, true/false statements) before moving into computation. Working through those conceptual questions first is a fast, honest way to check whether a section’s ideas actually landed before grinding through the longer computational problems.

Take advantage of the book’s own study aids where your PDF viewer supports them: Version 4.0 added 3D-interactive graphics throughout the Calculus III volume, viewable in Adobe Acrobat/Reader on desktop — click a 3D figure to rotate, zoom, and pan it, which is genuinely useful for building intuition about surfaces and solids in Chapters 12 and 13 that a flat, static picture in most textbooks cannot give you.

Used In These Programs

This book is used for the Calculus I / II / III course sequence in: BS Mathematics, and BS Engineering programs that require a full three-semester calculus sequence. Browse all Calculus books or all Mathematics books.

Who Should Read This

APEX Calculus suits a BS Math or BS Engineering student working through the standard three-semester Calculus I–III sequence, and it works equally well downloaded one volume at a time as your course reaches each part — a student only in Calculus I never needs to download Calculus II or III at all. It assumes a solid college-algebra and trigonometry background going in (functions, graphs, trig identities); a student who has not completed a college algebra course first will struggle with Chapter 1’s early examples, which lean on trig functions and algebraic manipulation without re-teaching them. It is not the right choice for a course that specifically requires a dedicated differential-equations unit or Lagrange multipliers in the multivariable chapter, since Version 4.0 covers neither. Students who want the same three-course syllabus with a more conventional, exercise-bank-heavy presentation and a slightly more formal tone can compare it directly against Calculus Volume 1 by Strang & Herman (OpenStax) — both cover the same three-semester ground, so the better fit usually comes down to which explanatory style and which set of practice problems a given student learns faster from.

Applicable Universities

This book is useful for students at Pakistani universities offering BS Mathematics and BS Engineering programs including Punjab University, Virtual University, COMSATS, FAST, UET, NUST, and other HEC-recognized institutions.

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Related Books

• Calculus Volume 1 – Strang & Herman (OpenStax)
• Active Calculus – Matthew Boelkins
• Calculus Volume 2 – Strang & Herman (OpenStax)
• Calculus Volume 3 – Strang & Herman (OpenStax)

APEX Calculus is a free, complete alternative to a paid calculus sequence for BS Math and BS Engineering students, built around real-world applications and, in its Calculus III volume, interactive 3D graphics most textbooks don’t offer at any price. Browse more Mathematics books for the rest of your semester.

APEX Calculus, by Gregory Hartman and contributors, Virginia Military Institute / APEX (Affordable Print and Electronic teXts). Free under the Creative Commons Attribution-NonCommercial 4.0 International licence. Official source: https://apexcalculus.com