MATH 2081 - Multivariable Calculus

Credits: 5
Hours/Week: Lecture None Lab None
Course Description: This is a first course in Multivariable Calculus. Topics include vectors in 3D-space, vector functions, functions of two or more variables, partial derivatives, gradients, and the chain rule; applications to max/min problems, Lagrange multipliers; double and triple integrals, change of variable, polar and spherical coordinates; line and surface integrals, vector fields and the fundamental theorem of line integrals; curl and divergence, theorems of Green and Stokes, and the Divergence theorem. Use of a 3-D graphing calculator, such as a TI-Nspire CX CAS, is required. Limited use of a computer algebra system will be made. This course is intended for students majoring in chemistry, engineering, physics, science, mathematics, mathematics education, or computer science.
MnTC Goals
None

Prerequisite(s): MATH 1082  with a grade of C or higher, or instructor consent.
Corequisite(s): None
Recommendation: Eligible for college-level Reading and English.

Major Content
1. Vectors

a.   Introduction to vectors

b.   Vectors in the plane and basic operations

c.   Vectors in space and basic operations

d.  Dot product and cross product

e.  Lines, planes, spheres, and quadric surfaces

f.  Vector-valued functions, space curves, and tangent vectors

g.  Differentiation and integration of vector functions

h.   Applications, including arc length and curvature

2. Differential Calculus of Functions of Several Variables

a.   Definition of a function of more than one variable

b.   Limits and continuity

c.   Partial derivatives

d.  Tangent planes and linear approximation

e.  Gradients and directional derivatives

f.   Applications to max/min problems

g.  Lagrange multipliers

3. Multiple Integrals

a.   Evaluating double and triple integrals

b.   Iterated integrals and change of order

c.   Applications

      i. Area

      ii. Volume

      iii. Mass and centroid

      iv. Probability

d.   Change of variable procedures

      i. Polar coordinates

      ii. Cylindrical coordinates

      iii. Spherical coordinates

      iv. Transformations and Jacobians

4. Vector Field Analysis

a.  Vector fields

b.  Line integrals and work

c.  Conservative vector fields

d.  Green’s theorem

e.  Parameterized surfaces

f.   Surface integrals

g.  Fundamental theorem of line integrals

h.  Divergence theorem

i.   Stokes’ theorem
Learning Outcomes
At the end of this course students will be able to:

1. demonstrate critical and logical reasoning when solving problems.
2. perform multi-dimensional vector operations such as vector addition, scalar multiplication, dot product, cross-product, magnitude, vector projection, and the angle between two vectors.
3. determine the multi-dimensional equations of lines, planes, spheres, and quadric surfaces.
4. explain the concepts of limits and continuity for real-valued functions of two or more variables.
5. find derivatives of vector-valued functions and use those derivatives to describe an object’s motion.
6. find the partial derivatives of a function of two or more variables.
7. analyze graphs of surfaces in rectangular, cylindrical, and spherical coordinates and in parameterized form.
8. find the tangent plane to the graph of a function.
9. determine any extreme values and saddle points of a function of several variables using partial derivatives and Lagrange multipliers.
10. compute gradients and directional derivatives and apply them to finding tangent spaces and normal lines.
11. evaluate iterated integrals using rectangular, cylindrical, and spherical coordinate systems.
12. solve problems such as calculating volume, center of mass, moments of inertia, and the expected value of a continuous random variable using double and triple integrals.
13. recognize and analyze vector fields, computing and interpreting curl, divergence, and flux.
14. calculate work done by a force field in moving an object along a curve using line integrals
15. state and apply the Fundamental Theorem of Line Integrals, Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem.
16. compare and contrast the generalizations of the Fundamental Theorem of Calculus listed above.
17. communicate clearly a problem’s solution and its explanation for the intended audience in terms of the problem posed.

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