GaussWeber Monument 



Wilhelm Weber (1804–1891) and Carl Friedrich Gauss (1777–1855)
GaussWeber Monument in Göttingen
Date 1899, by Carl Ferdinand Hartzer

Please bookmark this page, http://aleph0.clarku.edu/~djoyce/ma131/, so you can readily access it.
General information
 General description.
Multivariate calculus uses linear algebra to extend the important concepts
of singlevariable calculus to higherdimensional settings. Topics include
scalarvalued and vectorvalued functions, graphs, level sets, limits and
continuity; partial derivatives, gradients, tangent planes, differentiability,
total derivatives, directional derivatives; paths, velocity, acceleration,
arclength, curvature, vector fields, divergence, curl; extrema, Hessians,
Lagrange multipliers; multiple integrals, change of variables, Jacobians;
line integrals, Green’s theorem; surface integrals, Stokes’ theorem, and
Gauss’ theorem.
See also
Clark’s Academic Catalog.
 Course prerequisite : Math 122, 125, or 130, others by permission only.
 Web pages for related courses
Math 120, Calculus I
Math 121, Calculus II
Math 130, Linear Algebra
Math 217, Probability and Statistics
 Course Hours. MWF 10:00–10:50,
 Assignments & tests.
There will be numerous short assignments, mostly from the text, occasional quizzes,
two tests during the semester, and a twohour final exam during finals week.
The two tests during the semester are yet to be scheduled.
 Course grade. The course grade will be determined as follows:
2/9 assignments and quizzes,
2/9 each of the two midterms, and
1/3 for the final exam.
 Course goals.
 To provide students with a good understanding of the concepts and methods of multivariate
calculus, described in detail in the syllabus.
 To help the students develop the ability to solve problems using multivariate calculus.
 To connect multivariate calculus to other fields both within and without mathematics.
 To develop abstract and critical reasoning by studying proofs as applied to multivariate
calculus.

Textbook. The text for the course is
Vector Calculus, fourth edition, by Susan J. Colley, 2012.
http://www.pearsonhighered.com/educator/product/VectorCalculus/9780321780652.page
This is an excellent text. It includes a lot of extra material that you will find useful in your further studies. Keep it after the course as a good reference book.
Syllabus
Not all of the topics listed below will be covered in the same depth. Some are fundamental and will be covered in detail; others indicate further directions of study and will be treated as surveys. The only concepts of physics we will study in depth are velocity, acceleration, angular velocity, and angular acceleration, but a few others will be mentioned such as force.
There are probably more topics than we can discuss in one semester, so some will have to be eliminated because of time. Likely candidates are those in brackets.
The listed exercises are tentative. They may be changed as the course progresses.
1. Vectors 4 meetings
Much of chapter 1 is a review of material you’ve already seen
in Math 122 or in Math 130, but some of it will be new.
§ 1.1 Vectors in two and three dimensions
 R^{2}, R^{3}, vector notation, scalars
 Vector addition, zero vector, scalar multiplication, and their properties
 Geometric interpretation of vectors, parallelogram law for addition
 Exercises: 1–11 odd, 15, 23, 25
§ 1.2 Vectors and equations
 Standard basis vectors i, j, k
 Parametric equations for lines
 Symmetric form equations for a line in R^{3}
 Parametric equations for curves
x : R → R^{2}
 Velocity, speed, and acceleration
 Exercises: 1–7 odd, 13, 15, 17, 35, 44
§ 1.3 Dot products
 Dot products of vectors, lengths of vectors, law of cosines and angles
 Projections of vectors
 Normalization of vectors
 Exercises: 1–13 odd, 17, 21, 29, 30
§ 1.4 Cross products
 Cross products of pairs of vectors in R^{3}
 Areas of parallelograms and triangles
 Matrices, determinants
 Triple scalar product, volume of a parallelepiped
 Rotation, angular velocity
 Exercises: 1, 3, 5, 11–19 odd, 25
§ 1.5 Equations and distance
 Equations of planes, parametric equations of planes
 Distance between a point and a line
 Distance between parallel planes
 Distance between skew lines
 Exercises: 1, 3, 5, 13, 23, 25, 27, 31
§ 1.6 Summary of geometry in R^{n}
 Cauchy inequality, triangle inequality
 Standard basis
 Linear functions correspond to matrices, and composition of functions corresponds
to matrix multiplication
 Hyperplanes
 Determinants, minors, and cofactors
 Exercises: 3, 5, 7, 16–19, 21–23, 25,
27, 28a, 30a
§ 1.7 Coordinate systems
 Rectangular coordinates
 Polar coordinates for planes
 Cylindrical and spherical coordinates for space
 Exercises: 1–8, 11, 12, 15–17
2. Differentiation in several variables 8 meetings
You’ll see how concepts of limit, continuity, and derivatives generalize from the
onevariable case you saw in firstyear calculus to many variables.
§ 2.1 Functions of several variables
 Functions f : X → Y,
domain X, codomain Y, range R(f)
 Onto (surjective and onetoone (injective) functions
 Onetoone correspondences (bijective functions) and their inverses
 Scalarvalued functions
f : R^{n} → R,
also called scalar fields
 Vectorvalued functions
f : R^{n} → R^{m},
and their component functions
f_{i} : R → R^{m}
 Graphs of functions
R^{2} → R as surfaces in
R^{3}, and their level curves
 Surfaces in R^{3}, hypersurfaces in R^{n}
 [Quadric surfaces]
 Exercises: 1–7, 10, 15–21 odd, 31, 39
§ 2.2 Limits of vectorvalued functions
 Intuitive concept and formal definition of limits for multivariate functions
 Topological concepts: open and closed subsets, boundaries of subsets, neighborhoods of points
 Properties of limits
 Multivariate polynomials
 Continuous functions
 Exercises: 7–13, 29, 30, 38–43, 47, 48
§ 2.3 Derivatives for multivariate functions
 Partial derivatives for scalarvalued functions (scalar fields)
 Differentiability and planes tangent to surface graphs of functions
R^{2} → R
 Differentiability and hyperplanes tangent to hypersurface graphs of functions
R^{n} → R
 Differentiability for vectorvalued functions
 Gradient vectors for scalarvalued functions
 Derivative matrix for vectorvalued functions
 Exercises: 1–7, 12–14, 19–21, 29, 30, 34–36
§ 2.4 Properties of derivatives, higherorder partial derivatives
 Linearity: sum, difference, and constant multiple rules for vectorvalued functions
R^{n} → R^{m}
 Product and quotient rules for scalarvalued functions R^{n} → R
 Partial derivatives of higher order
 Exercises: 1, 2, 9–11, 20, 28, 29a
§ 2.5 The chain rule in several variables
 The chain rule for composition fog where
g : R → R^{n} and
f : R^{n} → R
 The chain rule for the composition
fog where
g : R^{n} → R^{m} and
f : R^{m} → R^{p}
 Polarrectangular conversions
 Exercises: 1, 2, 5, 8, 11, 15, 19, 22, 23
§ 2.6 Directional derivatives and gradients, implicit & inverse function theorems
 The gradient vector field for a scalar field
 Directional derivatives, definition and evaluation in terms of the gradient
 Steepest ascent
 Tangent planes and hyperplanes
 Exercises: 2, 3, 12, 13, 15, 16
§ 2.7 [Newton’s method]
3. Vectorvalued functions 5 meetings
§ 3.1 Parameterized curves; Kepler’s laws of planetary motion
 Functions x : R → R^{n}
as paths or parameterized curves
 Velocity, speed, and acceleration
 Kepler’s laws of planetary motion
 Exercises: 1–4, 7, 8, 15, 16
§ 3.2 Arc length, curvature
 Length of a path as an integral
 Unit tangent vector, curvature of a path or curve
 Exercises: 1, 2, 4, 10, 16, 22a
§ 3.3 Vector fields
 Functions f : R^{n} → R as scalar fields
 Functions F : R^{n} → R^{n}
as vector fields
 The gradient field (a vector field) associated to a potential function (a scalar field)
 Equipotential sets
 Flow lines of vector fields
 Exercises: 1, 4, 9, 10, 19–21, 24, 26
§ 3.4 Gradient, divergence, curl, the del operator
 The del operators and gradients
 The del operator and divergence of a vector field
 The curl of a vector field in R^{3}
 Gradient fields are irrotational, that is, curl(grad f) = 0
 Div(curl F) = 0
 Exercises: 1–4, 7–10, 13, 28a
4. Maxima and minima in several variables 4 meetings
§ 4.1 Differentials and Taylor’s theorem
 Taylor’s theorem for single variable functions as an extension of
the mean value theorem
 Taylor polynomials, remainder term
 The firstorder formula for the multivariate Taylor’s theorem
 Total differentials
 The secondorder formula and the Hessian
 Higherorder Taylor polynomials
 Exercises: 1, 2, 8, 9, 11, 19, 20, 24
§ 4.2 Maxima and minima (extrema) of functions
 Local minima and maxima for scalar fields
 Critical points and the Hessian criterion
 Quadratic forms, positive definite forms
 The second derivative test for extrema of scalarvalued functions
 Compact sets, the Extreme Value Theorem (EVT)
 Exercises: 3–6, 13–16
§ 4.3 Lagrange multipliers
 Equational constraints
 Lagrange multipliers for extrema subject to constraints
 Exercises: 3, 4, 5, 9
§ 4.4 Applications of extrema
 [Least squares approximation]
 [Applications to economics]
5. Multiple integration 6 meetings
§ 5.1 Areas and volumes
 Volumes over rectangles as double integrals
 Exercises: 1, 2, 3, 6, 9
§ 5.2 Double integrals
 Double integrals over rectangles defined as Riemann sums, integrability
 Conditions for integrability
 Fubini’s theorem, linearity of integrals, other basic properties
 Double integrals over general regions
 Exercises: 5–7, 10, 16
§ 5.3 Changing order of integration
§ 5.4 Triple integrals
 Triple integrals over boxes
 Properties of triple integrals
 Triple integrals over general regions
 Exercises: 1, 2, 5, 6, 11, 13, 17
§ 5.5 Change of variables
 Transformations of the plane
R^{2} → R^{2}
 Linear transformations and their expansion factors
 Change of variables in definite integrals of one variable
 Change of variables for double integrals, the Jacobian
 Double integrals in polar coordinates
 Change of variables for triple integrals
 Exercises: 1, 3, 9, 13, 17
§ 5.6 Applications of multiple integration
 [Mean value (average value) of a scalarvalued function]
 [Center of gravity]
6. Line integrals 5 meetings
§ 6.1 Scalar and vector line integrals
 Scalar line integrals
 Vector line integrals
 Reparameterization
 Exercises: 1–3, 9, 16, 17, 34
§ 6.2 Green’s theorem
 Green’s theorem
 Divergence theorem in the plane
 Exercises: 1–3, 7, 9, 15, 17
§ 6.3 Conservative vector fields
 Vector fields with pathindependent line integrals
 Gradient fields and line integrals, conservative vector fields
 Exercises: 3–6
7. Surface integrals 5 meetings
§ 7.1 Parameterized surfaces
 Coordinate curves, normal vectors, tangent planes
 Smooth and piecewise smooth surfaces
 Areas of surfaces
 Exercises: 1, 3, 24, 26
§ 7.2 Surface integrals
 Scalar surface integrals
 Vector surface integrals
 Reparameterization of surfaces
 Exercises: 1, 3, 7, 11
§ 7.3 Stokes’ and Gauss’ theorems
 Stokes’ theorem
 Gauss’ theorem
 Divergence and curl
 Exercises: 1, 3, 7, 9
Class notes, quizzes, tests, homework assignments
The dates for the discussion topics and the assignments are tentative. They will change as the course progresses.
 Monday, 13 Jan 2014. Welcome to the class! Outline of the course.
Things you need to know about linear algebra
Preview. We’ll study functions
R^{n} → R^{m}
where not both n and m are 1. Three important kinds of these.
 When m = 1,
f : R^{n} → R,
is a scalarvalued function or a scalar field.
 When n = 1,
x : R → R^{n} parameterizes a curve in
nspace, that is, the path of a moving point.
 When m = n,
F : R^{n} → R^{n}
describes a vector field on R^{n}.
 Wednesday, 15 Jan.
Notes on Curves and paths. Velocity, speed,
and acceleration
Quick review of important topics in linear algebra.
Discuss exercises from sections
1.1: Exercises: 1–11 odd, 15, 23, 25,
and
1.2. Exercises: 1–7 odd, 13, 15, 17, 35, 44.
Answers to section 1.2.
 Friday, 17 Jan.
More topics from linear algebra including dot products
and cross products.
Dot and cross products
Discuss exercises from sections
1.3: Exercises: 1–13 odd, 17, 21, 29, 30.
Answers to section 1.3.
Review Sets and set notation when we have time.
Monday, 20 Jan. Martin Luther King day. No classes.
 Wednesday, 22 Jan.
Discuss functions of several variables including concepts of domain,
codomain, range; onto (surjective) and onetoone (injective) functions,
onetoone correspondences (bijective functions) and their inverses;
scalarvalued functions (also called scalar fields), vectorvalued functions
and their component functions.
Notes on Functions.
Homework due from sections
1.4: Exercises: 1, 3, 5, 11–19 odd, 25.
Answers to section 1.4
Discuss section 1.5: Exercises: 1, 3, 5, 13, 23, 25, 27, 31.
 Friday, 24 Jan.
Graphs of functions as surfaces and
described by level curves; surfaces and hypersurfaces
Limits of functions of several variables including
the intuitive concept and formal definition of limits for multivariate functions;
topological concepts of open and closed subsets, boundaries of subsets,
neighborhoods of points
A gallery of surfaces
Notes on Limits.
Homework due from sections 1.5 (see above).
Answers to section 1.5.
Discuss section 1.6: Exercises: 3, 5, 7, 16–19, 21–23, 25, 27, 28a, 30a.
 Monday, 27 Jan.
Properties of limits; multivariate polynomials; continuous functions.
Partial derivatives
Notes on Partial derivatives
Exercises due from section 1.6 (see above).
Answers to section 1.6
 Wednesday, 29 Jan.
Tangent planes, total derivatives, gradients for scalar fields
R^{n} → R
Notes on Gradients
Exercises due from
section 2.1: 1–7, 10, 15–21 odd, 31, 39.
Answers
 Friday, 31 Jan.
Notes on total derivatives
Derivative matrix for vectorvalued functions
R^{n} → R^{m},
rules of differentiation
Exercises due from section 2.2:
7–13, 29, 30, 38–43, 47, 48.
Answers
 Monday, 3 Feb.
Quiz on dot and cross products.
Answers
Higher order derivatives
Discuss Exercises from section 2.3:
1–10, 15–17, 21, 22, 26–28 due Wed.
 Wednesday, 5 Feb. Snow day. No classes.
 Friday, 7 Feb.
Newton’s method
Chain rule, part 1
Exercises from section 2.3 due, see above.
Answers to section 2.3
Discuss Exercises due from section 2.4:
1, 2, 9–11, 16, 20, 21a due Monday.
 Monday, 10 Feb.
Chain rule, part 2
Exercises from section 2.4 due, see above.
Answers
Discuss Exercises from section 2.5:
1, 2, 5, 8, 9, 11, 15–17
 Wednesday, 12 Feb.
Directional derivatives, steepest ascent, tangent planes
Exercises from section 2.5 due, see above.
Answers
Discuss Exercises due Friday from section 2.6:
2, 3, 12, 13, 15, 16
 Friday, 14 Feb.
Speed, velocity, acceleration
Exercises from section 2.6 due, see above.
Answers
 Monday, 17 Feb. Review.
Discuss Exercises from section 3.1: 1–4, 7, 8, 15, 16 due Monday
 Wednesday, 19 Feb. First midterm on chapter 2.
First test. Answers.
 Friday, 21 Feb.
Arclength, the arclength parameter, and parameterizing curves by their arclength
Kepler’s laws
The arc length parameter s
 Monday, 24 Feb.
The unit tangent vector and curvature
Exercises from section 3.1: due, see above
Answers
 Wednesday, 26 Feb.
Vector fields, gradient fields, potential functions, equipotential sets, flow lines
Notes on Vector fields
Exercises due from section 3.2: 1, 2, 4, 8, 12, 18a.
Answers
 Friday, 28 Feb.
Critical points, maxes, mins, saddle points,
Notes on Critical points
Exercises due from section 3.3:
1, 4, 9, 10, 19–21, 24, 26.
Answers
Monday–Friday, 3–7 Mar. Spring Break
 Monday, 10 Mar.
The del operator, divergence of vector fields, incompressible vector fields,
curl of a vector field, irrotational vector fields
Notes on Gradient, divergence, and curl
A Lorenz attractor
Discuss Exercises due Wednesday from section 3.4:
14, 710, 13, 28a.
 Wednesday, 12 Mar.
Taylor's theorem for a single variable, Taylor polynomials, remainder term;
the firstorder formula for the multivariate Taylor's theorem, total differentials;
the secondorder formula and the Hessian.
Notes on Differentials and Taylor polynomials
Exercises from section 3.4 due, see above
Answers
Discuss Exercises due Friday from section 4.1:
1, 2, 8, 9, 11, 19, 20, 24
 Friday, 14 Mar.
Local minima and maxima for scalar fields, critical points and the
Hessian criterion; quadratic forms, positive definite form; the second
derivative test
Compact sets, the Extreme Value Theorem (EVT)
Maxima and minima of scalar fields
Exercises from section 4.1 due, see above.
Answers
 Monday, 17 Mar.
Quiz on sections 3.2–3.4.
Discuss Exercises due Wednesday from section 4.2:
36, 1316
Lagrange multipliers
 Wednesday, 19 Mar.
An application of Lagrange multipliers to pyramids
Exercises from section 4.2 due, see above.
Answers
Discuss Exercises due Friday from section 4.3:
3, 4, 5, 9.
 Friday, 21 Mar.
Double integrals. Volumes as integrals, integration
over rectangles and other regions, Riemann sums, integrability.
Exercises due from section 4.3 due, see above.
Answers
Discuss Exercises due Monday from section 5.1 and 5.2
 Monday, 24 Mar.
Fubini’s theorem, linearity of integrals, other basic properties.
Double integrals over general regions. Changing order of integration.
Triple integrals over boxes.
Exercises due from section 5.1: 1, 2, 3, 6, 9,
Answers
and from section 5.2: 5, 6, 7, 10, 16.
Answers
 Wednesday, 26 Mar.
Jacobians for change of variables.
Transformations of the plane
R^{2} → R^{2}
and their expansion factors. Change of variables and the Jacobian for double and
triple integrals
Double integrals in polar coordinates
Exercises due from section 5.3: 3–6, 15, 17,
Answers
 Friday, 28 Mar.
Exercises due from section 5.4: 1, 2, 5, 6, 11, 13, 17
Answers
 Monday, 31 Mar.
Review.
 Wednesday, 2 Apr. Second midterm on chapters 3–5.
Second test. Answers.
 Friday, 4 Apr.
Introduction to line integrals: scalar line integrals, vector line integrals,
and differential forms.
Line integrals
 Monday, 7 Apr.
Green's theorem as a generalization of the fundamental theorem of calculus,
Stokes' theorem and the divergence theorem in the plane.
Green’s theorem
 Wednesday, 9 Apr. Spree day. No classes.
 Friday, 11 Apr.
Proof of Green’s theorem
Exercises from section 5.5: 1, 3, 9, 13, 17 due
Answers
 Monday, 14 Apr.
Irrotational vector fields with simply connected domains are conservative.
Exercises from section 6.1.:
1–3, 9, 16, 17, 34 due
Answers
 Wednesday, 16 Apr.
Conservative vector fields:
as gradient fields, as vector fields having path
independent line integrals, as fields with 0 integrals over closed paths.
Exercises from section 6.2:
1–3, 7, 9, 15, 17 due.
Answers
 Friday, 18 Apr.
Surfaces in space, their parameterizations and
tangent planes.
Exercises from section 6.3: 3–6 due.
Answers
 Monday, 21 Apr.
Scalar surface integrals
Vector surface integrals
Exercises from section 7.1: 1, 3, 24, 26 due.
Answers
 Wednesday, 23 Apr. Academic Spree Day.
More on Stokes' theorem, orientation of surfaces,
the statement of the divergence theorem, a.k.a. Gauss's theorem.
Exercises from section 7.2: 1, 3, 7, 11 due.
Answers
 Friday, 25 Apr.
Gauss’s theorem
Exercises from section 7.3: 1, 3, 7, 9
due Monday.
 Monday, 28 Apr. Review
Answers to section 7.3 exercises
 Wednesday, 30 Apr. Reading day. Review
 Tuesday, 6 May. Final Exam. 8:00–10:00.
Past tests
This page is located on the web at
http://aleph0.clarku.edu/~djoyce/ma131/
David E. Joyce,