# ModelCourses/Multivariate Calculus

## Contents |

# Multivariate Calculus Model Course Units

A user of this material should locate appropriate units below that fit their particular course in multivariate calculus.

Instructions for importing problem sets Instructions for exporting problem sets

Within each Unit below, specific problem types should be described. Detailed problem descriptions are given by clicking on the unit title.

Complete problem sets for each unit will eventually be collected and made available from this site (and perhaps from within the WebWork system itself) but these have not been made available yet. Also, the specific problems suggested could be directly linked if desired although this might be a bit too much!

## Download this entire course of problem sets

## Vectors in Space

Download this entire collection of problem sets

### Unit 1 - Vector Algebra

- Vector Algebra
- Dot Product and Applications
- Cross Product and Applications

### Unit 2 - Lines and Planes

- Lines in Space
- Two forms of a line in space: (1) parametric equations and symmetric equations
- Express a line that passes two given points.
- Express a line that passes through a given point and is parallel to a given line.
- Express a line that passes through a given point and is perpendicular to a given line.
- Express a line that passes through a given point and is perpendicular to two given lines that are not parallel.
- Determine if two given lines are parallel, perpendicular or neither.
- Determine if two given lines intersect or are skew, and find the intersection if they intersect.
- Determine if a given line passes through a given point.

- Two forms of a line in space: (1) parametric equations and symmetric equations

- Planes in Space
- Plane formula
- Determine if a given plane contains a given point.
- Express a plane that contains a given point and is perpendicular to a given vector.
- Express a plane that contains a given point and is parallel to a given plane.
- Express a plane that contains a given point and is perpendicular to a given plane.
- Express a plane that contains a given point and is perpendicular to given two planes that are not parallel.
- Determine if two given planes are parallel, perpendicular or neither.
- Find the intersection of two planes that are not parallel.

- Plane formula

### Unit 3 - Cylindrical Surfaces and Quadric surfaces

- Cylindrical Surfaces
- Sketch the curve f(x,y)=0, z=z0 (or g(x,z)=0, y=y0 or h(y,z), x=x0) in the xyz-space.
- Sketch the surface f(x,y)=0 (or g(x,z)=0, or h(y,z)=0) in the xyz-space.

- Quadric Surfaces
- Quadratic equations for (1) sphere (2) ellipsoid (3) paraboloid (4) elliptic cone (5) hyperboloid of one sheet (6) hyperboloid of two sheet (7) hyperbolic paraboloid

### Unit 4 - Non-rectangular Coordinate Systems

- Cylindrical Coordinates
- Spherical Coordinates
- Applications

## Vector-valued Functions in One Variable

### Unit 1 - Calculus of Vector Functions

- Calculus of Vector-valued Function in One Variable
- Vector-value function r(t)
- Express the domain of r(t) a 2-D (or 3-D) vector-valued function in an interval notation.
- Determine if a given point is on the curve traced out by r(t).
- Sketch the curved traced out by r(t).
- Find all values of t at which r(t) is parallel to a given vector.
- Find all values of t at which r(t) is perpendicular to a given vector

- Derivatives of Vector Functions
- Compute the tangent vector r'(t) of r(t).
- Find all values of t at which r(t) and r'(t) are perpendicular.
- Show that r(t) and r'(t) are perpendicular at every t if the magnitude of r(t) is a constant.

- Integrals of Vector Functions
- Compute an indefinite integral of r(t).
- Compute a definite integral of r(t).

- Vector-value function r(t)

### Unit 2 - Vector Function Applications

- Arc Length
- Curvature
- Unit Tangent and Unit Normal vectors
- Computing the radius, center and equation of osculating circle
- Motion in Space: Velocity and Acceleration

## Multi-variable Functions

### Unit 1 - Multi-variable Functions

- Domain and range
- Find and sketch the domain of a given function.
- Evaluate a function at a given point.
- Find the range of a given function.
- Sketch level curves (surfaces) of f(x,y) (f(x,y,z)).
- Sketch the surface z=f(x,y) and traces f(x,y)=z0 on the surface.

- Limits
- Definition of limit
- Show the limit of a function does not exist by showing the function has two different limits along two distinct paths, respectively.
- Compute the limit or show limit does not exist using polar coordinates.
- Compute the limit or show limit does not exist using the definition of limit.

- Definition of limit
- Continuity
- Definition of continuity
- Determine if a function is continuous at a given point by checking if the limit is the same as the value of the function at this point.
- Find the region over which the function is continuous.

- Definition of continuity

### Unit 2 - Partial Derivatives - Rules

- Partial Derivatives
- Definitions of Partial Derivatives
- Compute the first partial derivative of f with respect to x (y or z).
- Sketch the tangent line at a given point on the trace f(x,y)=z0.
- Compute higher order partial derivatives.

- Definitions of Partial Derivatives

- Partial Derivatives using Rules
- Differentiation Rule
- The Chain Rule
- Implicit differentiation using partial derivatives

- Directional Derivatives and the Gradient Vectors
- Gradient Vectors
- Compute the gradient vector of f at a given point.
- Compute the gradient vector of f at any point in its domain.
- Find all points at which the gradient of f is a zero vector.
- Sketch the gradient vector of f(x,y) at (x0,y0) on the level curve f(x,y)=f(x0,y0).

- Definition of Directional Derivative
- Directional Derivative as the dot product of gradient vector and unit direction vector
- Compute the directional derivative of f at a point along a unit direction.
- Compute the directional derivative of f as (x,y) (or (x,y,z)) moves from point A to point B.
- Find the maximum (minimum) rate of change of f at a given point and the corresponding direction vector.
- Determine if f is increasing, decreasing or neither at a given point along a give direction.

- Gradient Vectors

### Unit 3 - Partial Derivatives - Applications

- Tangent Planes and Linear and Quadratic Approximations
- Tangent Planes and Linear Approximations
- Find the equation of the tangent plane of the surface z=f(x,y) at (x0,y0).
- Find all points (x,y) at which the tangent plane of z=f(x,y) at (x0,y0) is parallel to a given plane.
- Find the linear approximation L(x,y) of f(x,y) at (x0,y0).
- Approximate f(x1,y1) by L(x1,y1).

- Quadratic Approximations
- Find the quadratic approximation Q(x,y) of f(x,y) at (x0,y0).
- Approximate f(x1,y1) by Q(x1,y1).

- Tangent Planes and Linear Approximations
- Maximum and Minimum Values
- Critical Points
- Find all critical points of f(x,y) (or f(x,y,z)).

- Second Derivative Test for Local Maximum and Minimum
- Find all local maximum and minimum of f(x,y) at the critical point (x0,y0) by the Second Derivative Test.

- Absolute Maximum and Minimum
- Find the absolute maximum and minimum of f(x,y) over a region.

- Critical Points
- Lagrange Multipliers

## Multiple Integrals

### Unit 1 - Double Integrals and Applications

- Double Integrals
- Double Integrals in the Rectangular Coordinates
- Set up and evaluate a double integral over a rectangular region.
- Set up and evaluate a double integral over a general region.
- Change the order of a given double integral.

- Double Integrals in Polar Coordinates
- Set up and evaluate a double integral over a region in polar coordinates.
- Express a given double integral in rectangular coordinates in polar coordinates.

- Double Integrals in the Rectangular Coordinates
- Applications of Double Integrals
- Area of a region
- Average value of f(x,y) over a region
- Surface area of a solid
- Volume of the solid under the surface z=f(x,y) over a region
- Centroid of a region, center of mass

### Unit 2 - Triple Integrals and Applications

- Triple Integrals
- Triple Integrals in Rectangular Coordinates
- Triple Integrals in Cylindrical Coordinates
- Triple Integrals in Spherical Coordinates
- Set up a triple integral for in the rectangular, cylindrical or spherical coordinates.
- Exchange the order of integration of a triple integral.
- Exchange the coordinate systems of a triple integral.

- Change of Variables in Multiple Integrals
- Applications of Triple Integrals
- Volume of a Solid
- Average Value of f(x,y,z) over a Solid
- Center of Mass

## Vector Calculus

### Unit 1 - Vector Fields

- Vector Fields
- Vector Fields in 2D and 3D
- Gradient Vector Fields, and Conservative Vector Fields
- For a given vector field, determine if it is conservative and find its potential function if it is conservative.
- Sketch a vector field at a given point.
- Sketch a trace of the flow of a velocity vector field.

### Unit 2 - Line Integrals in 2D and Applications

- Parametric Equations for Curves (line segments, circle, ellipses)
- Line Integrals of a scalar function
- Set up a line integral of f(x,y) over a curve C.
- Compute the total mass of a wire with a given density function.

- Line Integrals of a vector field
- Set up a line integral of vector field over a curve C.
- Compute the total work done by a given force vector field along a curve C.
- Compute the total circulation of a vector field along a closed curve C.

- The Fundamental Theorem of Calculus for Line Integrals
- Use a potential function of a conservative vector field F to evaluate the line integral of F along a curve C.

- Green's Theorem
- Use Green's Theorem to evaluate a line integral of a vector field over a closed curve when all required conditions are satisfied.

### Unit 3 - Line Integrals in 3D and Applications

- Parametric Surfaces and Areas (sometimes optional due to time constraints)
- Curl and Divergence (sometimes optional due to time constraints)
- Surface Integrals (sometimes optional due to time constraints)
- Stokes' Theorem (often optional)
- The Divergence Theorem (often optional)

# Packaged Courses

## Moodle

https://test.webwork.maa.org/moodle/

## Stewart

## Hughes-Hallett

## Smith and Minton

## Larson

# Other Model Course Pages

``Future Work: A rubric needs to be developed that helps instructors determine the hardness level of a particular problem.``

- Development Workgroup: Mei Qin Chen, Dick Lane and John Travis
- To Do:
- Finish choosing problem sets for remaining units
- Add features to problems to include:
- Hints
- Solutions
- MetaTags
- Improvements such as changing multiple choice problems to fill in the blank, etc.