General information

• General description. Multivariate calculus uses linear algebra to extend the important concepts of single-variable calculus to higher-dimensional settings. Topics include scalar-valued and vector-valued 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 the course description from Clark's Academic Catalog.

• Course prerequisite :  Math 122 or Math 130, others by permission only.

• Web pages for related courses
  Math 120, Calculus I
  Math 121, Calculus II
  Math 122, Calculus III
  Math 130, Linear Algebra
  Math 217, Probability and Statistics

• Course Hours. MWF ***.

• Office hours.

• Course Schedule

• Assignments & tests. There will be numerous short assignments, mostly from the text, occasional quizzes, two tests during the semester, and a two-hour 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, third edition, by Susan Jane Colley, Prentice-Hall, 2006. ISBN-10: 0131858742, ISBN-13: 9780131858749. It's been in print for a couple years, so you should be able to find it used for less than list price.
  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

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.

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
• R², R³, vector notation, scalars
• Vector addition, zero vector, scalar multiplication, and their properties
• Geometric interpretation of vectors, parallelogram law for addition

§ 1.2   Vectors and equations
• Standard basis vectors i, j, k
• Parametric equations for lines
• Symmetric form equations for a line in R³
• Parametric equations for curves x :  R → R²

§ 1.3   Dot products
• Dot products of vectors, lengths of vectors, law of cosines and angles
• Projections of vectors
• Normalization of vectors

§ 1.4   Cross products
• Cross products of pairs of vectors in R³
• Areas of parallelograms and triangles
• Matrices, determinants
• Triple scalar product, volume of a parallelepiped
• Rotation, angular velocity

§ 1.5   Equations and distance
• Equations of planes, parametric equations of planes
• Distance between a point and a line
• Distance betweeen parallel planes
• Distance between skew lines

§ 1.6   Summary of geometry in Rⁿ
• Cauchy inequality, triangle inequality
• Standard basis
• Linear functions correspond to matrices, and composition of functions corresponds to matrix multiplication
• Hyperplanes
• Determinants, minors, and cofactors

§ 1.7   Coordinate systems
• Rectangular coordinates
• Polar coordinates for planes
• Cylindrical and spherical coordinates for space

2. Differentiation in several variables       8 meetings

You'll see how concepts of limit, continuity, and derivatives generalize from the one-variable case you saw in first-year calculus to many variables.

§ 2.1   Functions of several variables
• Function f :  X → Y, domain X, codomain Y, range R(f)
• Onto (surjective and one-to-one (injective) functions
• One-to-one correspondences (bijective fuctions) and their inverses
• Scalar-valued functions f :  Rⁿ → R, also called scalar fields
• Vector-valued functions f : Rⁿ → R^m, and their component functions f_i : R → R^m
• Graphs of functions R² → R as surfaces in R³, and their level curves
• Surfaces in R³, hypersurfaces in Rⁿ
• [Quadric surfaces]

§ 2.2   Limits of vector-valued 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

§ 2.3   Derivatives for multivariate functions
• Partial derivatives for scalar-valued functions
• Differentiability and planes tangent to surface graphs of functions R² → R
• Differentiability and hyperplanes tangent to hypersurface graphs of functions Rⁿ → R
• Differentiability for vector-valued functions
• Gradient vectors for scalar-valued functions
• Derivative matrix for vector-valued functions

§ 2.4   Properties of derivatives, higher-order partial derivatives
• Linearity:  sum, difference, and constant multiple rules for vector-valued functions Rⁿ → R^m
• Product and quotient rules for scalar-valued functions Rⁿ → R
• Partial derivatives of higher order

§ 2.5   The chain rule in several variables
• The chain rule for composition fog where g : R → Rⁿ and f : Rⁿ → R
• The chain rule for the composition fog where g : Rⁿ → R^m and f : R^m → R^p
• Polar-rectangular conversions

§ 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

3. Vector-valued functions       5 meetings

§ 3.1   Parameterized curves; Kepler's laws of planetary motion
• Functions f : R → Rⁿ as paths or parameterized curves
• Velocity, speed, and acceleration
• Kepler's laws of planetary motion

§ 3.2   Arclength, curvature
• Length of a path as an integral
• Unit tangent vector, curvature of a path or curve

§ 3.3   Vector fields
• Functions f : Rⁿ → R as scalar fields
• Functions F : Rⁿ → Rⁿ as vector fields
• The gradient field (a vector field) associated to a potential function (a scalar field)
• Equipotential sets
• Flow lines of vector fields

§ 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 R³
• Gradient fields are irrotational, that is, curl(grad f) = 0
• Div(curl F) = 0

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 first-order formula for the multivariate Taylor's theorem
• Total differentials
• The second-order formula and the Hessian
• Higher-order Taylor polynomials

§ 4.2   Maxima and minima (extrema) of functions
• Local minima and maxima of scalar-valued functions
• Critical points and the Hessian criterion
• Quadratic forms, positive definite forms
• The second derivative test for extrema of scalar-valued functions
• Compact sets, the Extreme Value Theorem (EVT)

§ 4.3   Lagrange multipliers
• Equational constraints
• Lagrange multipliers for extrema subject to constraints

5. Multiple integration       6 meetings

§ 5.1   Areas and volumes
• Volumes over rectangles as double integrals

§ 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

§ 5.3   Changing order of integration

§ 5.4   Triple integrals
• Triple integrals over boxes
• Properties of triple integrals
• Triple integrals over general regions

§ 5.5   Change of variables
• Transformations of the plane R² → R²
• 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

§ 5.6   Applications of multiple integration
• Mean value (average value) of a scalar-valued function
• Center of gravity

6. Line integrals       5 meetings

§ 6.1   Scalar and vector line integrals
• Scalar line integrals
• Vector line integrals
• Reparametrization

§ 6.2   Green's theorem
• Green's theorem
• Divergence theorem in the plane

§ 6.3   Conservative vector fields
• Vector fields with path-independent line integrals
• Gradient fields and line integrals, conservative vector fields

7. Surface integrals       5 meetings

§ 7.1   Parameterized surfaces
• Coordinate curves, normal vectors, tangent planes
• Smooth and piecewise smooth surfaces
• Areas of surfaces

§ 7.2   Surface integrals
• Scalar surface integrals
• Vector surface integrals
• Reparametrization of surfaces

§ 7.3   Stokes' and Gauss' theorems
• Stokes' theorem
• Gauss' theorem
• Divergence and curl

Sample tests

• Sample first test
• Sample second test
• Sample final exam

Class notes, quizzes, tests, homework assignments

1. Things you need to know about linear algebra

This page is located on the web at

http://aleph0.clarku.edu/~djoyce/ma131/

David E. Joyce,