CLP-3 Multivariable Calculus Study Guide and Chapter Summaries – Feldman, Rechnitzer & Yeager (2024 Version) BS Math/Engineering

BS Mathematics and BS Engineering students can use this guide to study CLP-3 Multivariable Calculus by Joel Feldman, Andrew Rechnitzer and Elyse Yeager (University of British Columbia). It is the third book in the CLP series and covers the usual Calculus III syllabus in three chapters: vectors and geometry in two and three dimensions, partial derivatives (including the gradient, optimisation and Lagrange multipliers) and multiple integrals in Cartesian, polar, cylindrical and spherical coordinates.

The book assumes single-variable calculus and keeps pointing back to the authors’ CLP-1 and CLP-2 texts, so it fits best after those two courses. It reads like a course text: it explains why each definition looks the way it does, builds each integral from a physical picture such as the mass of a plate, and works through about 180 numbered examples. Optional sections give proofs and extra applications, and eight appendices hold reference tables. One thing to know: the book is text only — it has worked examples but no end-of-section exercise sets, and line integrals, Green’s theorem and Stokes’ theorem belong to the next book, CLP-4.

Book Overview

Chapter List

Chapter 1: Vectors and Geometry in Two and Three Dimensions

Difficulty: Medium · Key topics: points and distance in 2d and 3d, vectors, dot product and projection, cross product, determinants, lines and planes, parametrised curves, tangent vectors and arc length, level surfaces, cylinders, quadric surfaces

The chapter builds the geometry that the rest of the book takes for granted. It starts with points, distance and the equation of a sphere, then defines vectors, their sums and scalar multiples, and the dot product, which gives angles, perpendicularity and projections. The cross product follows, introduced through the 2 by 2 determinant, the area of a parallelogram and the volume of a parallelepiped. Lines and planes are written from a point plus a direction or a normal vector. Next, vector-valued functions r(t) are differentiated to get tangent vectors, velocity and arc length. The chapter closes with sketching surfaces from cross-sections, level surfaces, cylinders and quadrics.

Key Points:

• The distance from (x, y, z) to the origin is √(x^2 + y^2 + z^2), so the sphere with centre (a, b, c) and radius R is (x-a)^2 + (y-b)^2 + (z-c)^2 = R^2.
• The dot product is a · b = a1 b1 + a2 b2 + a3 b3 = |a| |b| cos θ; it is zero exactly when the vectors are perpendicular (or one is the zero vector).
• The projection of a on b is the part of a pointing along b; its signed length is a · b / |b|.
• The cross product a × b is perpendicular to both vectors, has length |a| |b| sin θ (the area of their parallelogram), and obeys b × a = -(a × b).
• The volume of the parallelepiped with edges a, b, c is |a · (b × c)|, the absolute value of a 3 by 3 determinant.
• A plane with normal n = (A, B, C) through (x0, y0, z0) is A(x-x0) + B(y-y0) + C(z-z0) = 0; a line in 3d is (x, y, z) = (x0, y0, z0) + t d, and the intersection of two planes is a line.
• For a curve r(t), the derivative r′(t) is a tangent vector and its length |r′(t)| is the speed ds/dt, so arc length is the integral of |r′(t)|.

Memory Tip: The dot product answers “how much do these two point the same way?” and returns a number built from cos. The cross product answers “how far do they spread apart?” and returns a vector built from sin. If a · b = 0 they are perpendicular; if a × b = 0 they are parallel.

Common Mistake: Treating the cross product like ordinary multiplication. Swapping the order flips the sign, and a × (b × c) is not the same as (a × b) × c; the book singles these properties out as a frequent source of errors.

Important Questions:

• How do you find the angle between two vectors? Use cos θ = (a · b) / (|a| |b|). If the dot product is zero the angle is 90 degrees and the vectors are orthogonal.
• How do you write the equation of a plane from a point and a normal vector? Require n · (x - x0) = 0, which expands to A(x-x0) + B(y-y0) + C(z-z0) = 0. The coefficients of x, y, z in the plane’s equation are the components of a normal vector.

Chapter 2: Partial Derivatives

Difficulty: Hard · Key topics: limits and continuity in several variables, partial derivatives, mixed partials, chain rule, tangent planes and normal lines, linear and quadratic approximation, gradient and directional derivatives, partial differential equations, local and absolute extrema, Lagrange multipliers

This is the longest chapter and the heart of the book. It extends limits and continuity to several variables and shows why they are harder: a point can be approached along infinitely many paths. Partial derivatives follow, read as slopes of cross-sections, then higher-order derivatives and the equality of mixed partials. The chain rule leads to tangent planes, normal lines, and linear and quadratic approximations with error bounds. The gradient and directional derivatives show the direction of fastest increase. A short section checks solutions of partial differential equations. The chapter ends with local extrema, the second derivative test, absolute extrema on a region and Lagrange multipliers.

Key Points:

• A limit at (a, b) exists only if f(x, y) approaches the same value along every path; switching to polar coordinates and letting r → 0 is a standard way to test a limit at the origin.
• To find ∂f/∂x, treat y as a constant and differentiate as usual; it is the slope of the cross-section of the graph in the plane y = b. If both mixed partials are continuous, f_xy = f_yx (Clairaut).
• Chain rule: for F(s, t) = f(x(s, t), y(s, t)), ∂F/∂s = f_x ∂x/∂s + f_y ∂y/∂s, one term for each variable that f depends on.
• The tangent plane to z = f(x, y) at (x0, y0) is z = f(x0, y0) + f_x (x-x0) + f_y (y-y0); the same expression is the linear approximation of f near the point.
• The gradient ∇f has one component per variable. The directional derivative along a unit vector u is ∇f · u, f rises fastest along ∇f at rate |∇f|, and ∇f is perpendicular to level curves and surfaces.
• At a critical point (∇f = 0) compute D = f_xx f_yy - (f_xy)^2: D > 0 with f_xx > 0 is a local minimum, D > 0 with f_xx < 0 a local maximum, D < 0 a saddle point, and D = 0 settles nothing.
• Lagrange multipliers: at an extreme value of f on the constraint g = 0, ∇f = λ ∇g; solve this together with g = 0, then compare f at every candidate point.

Memory Tip: For the chain rule, draw the tree diagram: f at the top, x and y below it, then s and t below those. Multiply the partial derivatives along each path from f down to the variable you are differentiating by, then add up all the paths.

Common Mistake: Calling every critical point a maximum or minimum. ∇f = 0 only gives candidates, and the book works examples where the point is a saddle. For an absolute maximum on a closed region you must also check the boundary, not just the interior critical points.

Important Questions:

• In which direction does a function increase fastest, and how fast? Along the gradient ∇f(a, b), at rate |∇f(a, b)| per unit distance. It falls fastest in the opposite direction and stays constant along the level curve, which is perpendicular to the gradient.
• How does the second derivative test classify a critical point? Compute D = f_xx f_yy - (f_xy)^2 there. Positive D gives a minimum when f_xx > 0 and a maximum when f_xx < 0; negative D gives a saddle point; D = 0 is inconclusive.

Chapter 3: Multiple Integrals

Difficulty: Hard · Key topics: double integrals by vertical and horizontal slices, Fubini’s theorem, even and odd integrands, polar coordinates and polar curves, average value, centre of mass, moment of inertia, surface area, triple integrals, cylindrical and spherical coordinates, Jacobian (Section 3.8 optional)

The chapter builds the double integral from a physical question, the mass of a plate with variable density, by slicing the region, summing small rectangles and taking limits. Fubini’s theorem lets the integral be done as an iterated integral, and the book spends time on sketching the region and choosing the order. Polar coordinates handle round regions. Applications include average values, centre of mass, centroids, moment of inertia and the area of a surface z = f(x, y). Triple integrals follow in Cartesian, cylindrical and spherical coordinates. The optional last section derives the Jacobian rule for a general change of variables.

Key Points:

• A double integral is computed as an iterated integral: the inner limits may depend on the outer variable, the outer limits are plain numbers, and for a continuous integrand on a reasonable region Fubini’s theorem says either order gives the same value.
• ∬∬_R dA is the area of R; for f ≥ 0, ∬∬_R f dA is the volume under z = f(x, y); with f as a density it is the mass.
• If the integrand is odd under x → -x and the region is symmetric about the y-axis, the double integral is zero; the book gives the matching rules for the other reflections.
• Polar coordinates: x = r cos θ, y = r sin θ and dA = r dr dθ; they suit discs, sectors and curves such as the cardioid r = 1 + cos θ.
• The average of f over R is ∬∬ f dA divided by the area of R; the centre of mass is the average of x and y weighted by the density; moment of inertia uses the squared distance to the axis.
• The area of the surface z = f(x, y) over a region is ∬∬ √(1 + f_x^2 + f_y^2) dA.
• Volume elements: dV = r dr dθ dz in cylindrical coordinates and dV = ρ^2 sin φ dρ dφ dθ in spherical coordinates (φ measured from the positive z-axis); in general, dA picks up the Jacobian determinant.

Memory Tip: Set up limits from the outside in. The outermost integral has constant limits, each inner integral may use only the variables outside it, and the innermost limits may depend on all of them. Sketch the region, draw one typical slice, and read off where the slice starts and ends.

Common Mistake: Forgetting the extra factor after changing coordinates, such as writing dA = dr dθ instead of r dr dθ, or leaving out ρ^2 sin φ in spherical coordinates. The factor is the size of the tiny box in the new coordinates. Also check which symbol means which angle, because physics texts swap θ and φ (see Appendix G).

Important Questions:

• When can you swap the order of integration in a double integral? By Fubini’s theorem, when the integrand is continuous on a region that can be described both as vertical strips and as horizontal strips with continuous boundary curves; both orders then give the same value. The book warns that the hypotheses can be relaxed only a little, and shows an example with a single bad point where the two orders disagree.
• Why does a polar double integral contain the factor r? A small polar patch is almost a rectangle with sides dr and r dθ, so its area is r dr dθ. Patches further from the origin are wider for the same dθ.

Appendices A–H: Reference Material

Appendix · Difficulty: Easy · Key topics: trigonometry, powers and logarithms, tables of derivatives, integrals and Taylor expansions, Cartesian, cylindrical and spherical coordinates, ISO notation, conic sections and quadric surfaces

The eight appendices are reference pages that the chapters keep calling back to. Appendices A to E collect the single-variable toolkit from CLP-1 and CLP-2: trigonometry with special triangles, identities and inverse functions, powers and logarithms, and tables of derivatives, integrals and Taylor expansions. Appendix F gathers the definitions of Cartesian, cylindrical and spherical coordinates, with the surfaces of constant coordinate and the volume element for each. Appendix G repeats them in the ISO 80000-2 symbols used in physics and engineering, where the angles change roles. Appendix H lists the conic sections and the quadric surfaces with their cross-sections.

Key Points:

• Appendix F is the quickest place to check a coordinate conversion or a volume element: each of the three systems gets its definitions, its surfaces of constant coordinate and its dV.
• In the ISO convention of Appendix G, spherical coordinates use r for the distance, θ for the angle from the z-axis and φ for the angle around it; the main text uses ρ, φ and θ for the same three quantities.
• Appendices C, D and E are the tables the worked examples draw on: derivatives, integrals, and Taylor (Maclaurin) series of the standard functions.
• Appendix A’s trigonometry and Appendix B’s powers and logarithms are the algebra behind the trigonometric substitutions and the polar-coordinate integrals in Chapter 3.
• Appendix H describes conic sections as the curves cut from a cone by a plane, and lists every quadric surface in tables of its cross-sections, which extends the sketching practice of Chapter 1.

Memory Tip: Bookmark Appendix F before you start Chapter 3. The two volume elements you will look up most, r dr dθ dz and ρ^2 sin φ dρ dφ dθ, are both there next to a picture of the tiny box they measure.

Common Mistake: Copying a spherical-coordinate formula from a physics or engineering sheet into this book’s notation without checking Appendix G. The symbols for the two angles are swapped between the conventions, so the formula comes out with the wrong angle in it.

Important Questions:

• How does the ISO notation for spherical coordinates differ from the main text? ISO writes r for the distance, θ for the angle measured from the z-axis and φ for the angle in the xy-plane. The main text writes ρ, φ and θ for those three.
• What is a quadric surface? A surface given by a polynomial equation of degree two in x, y, z, such as a sphere, ellipsoid or hyperboloid. Its cross-sections are conic sections, so a quadric can be sketched by stacking ellipses, hyperbolas or parabolas.

How to Study This Book

Read the three chapters in order, because each leans on the one before: Chapter 1 supplies vectors, the dot and cross product and the language of surfaces, Chapter 2 uses them for gradients and tangent planes, and Chapter 3 uses both for surface area, volume elements and the Jacobian. The book also assumes single-variable calculus and constantly refers back to the authors’ CLP-1 and CLP-2 texts by theorem number, so before you start you should be comfortable with derivative rules, the chain rule, integration techniques including trigonometric substitution, and Taylor polynomials.

In Chapter 1, the core is Sections 1.1, 1.2.1, 1.2.2 and 1.2.5 (points, vector arithmetic, the dot product and the cross product), followed by lines and planes in Sections 1.3 to 1.5. The pendulum in 1.2.3, the identities in 1.2.6 and the rotating frames in 1.2.7 and 1.2.8 are optional applications. Section 1.2.4, on areas of parallelograms, is also marked optional but is worth reading if you have not met determinants, because it explains where the cross product formula comes from; the book says it uses matrices only as notation and needs no linear algebra. Section 1.6 on curves and tangent vectors is the bridge to vector calculus, and Sections 1.7 to 1.9 on sketching surfaces are skills you only gain by drawing, so redraw the book’s cross-section figures yourself.

Chapter 2 is where to spend most of your time. Read Sections 2.1 to 2.4 in order (limits, partial derivatives, higher-order derivatives, the chain rule); the proofs in 2.3.1, 2.3.2, 2.4.3 and 2.4.4 and the nasty limit in 2.1.1 can wait. Section 2.5 on tangent planes needs the gradient from Definition 2.5.4, and Section 2.6 on linear and quadratic approximation sets up the second derivative test, so do not skip 2.6.1. Section 2.7 (gradient and directional derivatives) is short but is used everywhere afterwards. Section 2.8 is a first look at partial differential equations and mostly checks that given functions satisfy them; 2.8.1 and 2.8.2 are optional.

Section 2.9 on maxima and minima runs about 33 pages and has the most worked examples in the book, including the second derivative test and the open-box problem, so work through it in two sittings. Its last part, 2.9.1, covers absolute maxima and minima on a region, which needs the boundary-checking procedure; read it before Section 2.10 on Lagrange multipliers, because the two methods answer the same kind of question. Treat 2.10.1 (two constraints) as optional unless your course uses it.

In Chapter 3, Sections 3.1.1 to 3.1.4 are the foundation: practise drawing the region and choosing vertical or horizontal slices until setting up limits feels automatic. Subsection 3.1.6 on even and odd integrands is a useful time-saver. Section 3.2 (polar coordinates), 3.3 (average value, centre of mass, moment of inertia) and 3.4 (surface area) can be read in order, and Sections 3.5 to 3.7 repeat the same slicing idea in three dimensions. Section 3.8 on general coordinates and the Jacobian is optional for most courses, but a BS Mathematics student heading towards analysis or probability should read it, since the Jacobian reappears whenever variables are changed in an integral.

What the book does not cover: line integrals, surface integrals, Green’s theorem, Stokes’ theorem and the divergence theorem. The official CLP page lists those in a fourth book, CLP-4 Vector Calculus, and the text itself points there. If your Calculus III course includes them, pair this book with Calculus Volume 3 from OpenStax, whose Chapter 6 covers vector calculus. Infinite sequences and series are also outside this book.

One thing to plan for: this PDF has about 180 numbered worked examples but no end-of-section exercise sets. The official CLP page describes the series as free textbooks and problem books, so check it for matching practice, or take exercises from another free multivariable text. A good routine is to cover each worked example, set up the problem yourself and compare your set-up (limits, coordinates, constraint) with the book’s before computing anything.

Used In These Programs

This book is used for the Multivariable Calculus (Calculus III) course in: BS Mathematics and BS Engineering programs. Browse all Multivariable Calculus books, all Calculus books or all Mathematics books.

Who Should Read This

CLP-3 suits a BS Mathematics or BS Engineering student taking Calculus III or multivariable calculus after a full single-variable sequence, particularly one who wants each formula built from a picture and explained in full sentences. It was written at the University of British Columbia as a course text, so it works well as the main reading for the vectors, partial derivatives, optimisation and multiple integrals part of the syllabus, and the optional sections give students heading toward analysis a little extra depth. It is not the right single choice for a student who needs a large bank of exercises, since the PDF has none, or for a course that also covers line and surface integrals and the big integral theorems, which belong to the next book in the series. A student who wants a second explanation with exercise sets can pair it with Calculus Volume 3, and a student who wants to see determinants, the quadric classification and the eigenvalue idea behind the second derivative test treated properly can read A First Course in Linear Algebra alongside it.

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.

Official Copy of CLP-3 Multivariable Calculus

This book is free from its official source. Click below to open the official CLP page at the University of British Columbia, where you can download the free PDF of CLP-3 Multivariable Calculus.

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Related Books

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CLP-3 Multivariable Calculus is a free, carefully explained Calculus III text from UBC for BS Mathematics and BS Engineering students, with about 180 worked examples on vectors, partial derivatives, optimisation and multiple integrals. Browse more Mathematics books for the rest of your semester.

CLP-3 Multivariable Calculus, by Joel Feldman, Andrew Rechnitzer and Elyse Yeager, University of British Columbia. Copyright 2017–2024 the authors. Free under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International licence. Official source: https://personal.math.ubc.ca/~CLP/