Math 50C- Multivariable Calculus
Activities
Instructor: David Arnold
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The Activities
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Vectors and Matrices in Matlab
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Plotting in Matlab
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Surfaces in Matlab
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Lines and Planes in Matlab
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Cylindrical Coordinates and Quadric Surfaces
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Curvature
-
Linear Approximations
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Parametric Equations
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Snell's Law
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Special Plane Curves
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An Introduction to Matlab Function M-Files
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Graphing Polar Equations in Matlab --- Script Files
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Polar Equations of Conic Sections
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Velocity and Acceleration---Constant Speed
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Level Curves in Matlab
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The Gradient
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Max-Min and Saddle Points
Abstracts and materials
Vectors and Matrices in Matlab
Abstract-
In this exercise you will learn how to enter vectors and matrices in Matlab.
Operations involving
vectors and matrices will be discussed.
Prerequisites-
Plotting in Matlab
Abstract-
In this activity you will learn how to plot lines and curves in both the
plane and space. In addition, you will also be introduced to script files.
Prerequisites-
-
Some knowledge of how to enter vectors and matrices in Matlab. Some familiarity
with Matlab’s array operations is useful.
Parametric Equations
Abstract-
This laboratory exercise is intended to introduce multivariable calculus
students to the Matlab software. Vectors are introduced and vector arithmetic
is practiced. Readers are shown how to draw the graphs of parametric equations
using Matlab's plot command.
Prerequisites-
Snell's Law
Abstract-
In this activity you will devise a proof of Snell's law of refraction based
on Fermat's Principle of Least Time.
Prerequisites-
-
The derivative, critical values, max and min, and the fact that the distance
traveled by a particle moving at constant speed is governed by the equation
d
= st
Special Plane Curves
Abstract-
We've discovered several parametrizations of the cycloid in class. There
are many beautiful curves similar to the cycloid, some of which were known
to the ancient greeks. In this assignment, you will have the opportunity
to choose and throughly examine a special plane curve of your choice.
Prerequisites-
-
Some experience finding the parametrization of a path. Some experience
plotting parametric equations using Matlab plot command.
Graphing Polar Equations in Matlab --- Script Files
Abstract-
Two techniques are used to graph polar equations of the form r=f(theta).
Matlab's polar command is introduced and utilized. An alternative
approach uses the polar to Cartesian transformations x=r cos(theta)
and y=r sin(theta) and Matlab's plot command. Matlab script
files enable students to organize and save their work for later use.
Prerequisites-
-
Familiarity with vector operations in Matlab, particularly Matlab's element-wise
operators. Some familiarity with Matlab's plot command is useful.
An Introduction to Matlab Function M-Files
Abstract-
In this activity readers will learn how to program rudimentary function
M-files. An example of a function M-file is used in a call to Matlab's
quad
command, a numerical integration routine.
Prerequisites-
-
Some introductiory knowledge of Matlab's vector structure, particularly
Matlab's element wise operations on vectors such as .* and .^
(See Matlab and vectors). Familiarity with saving files (See Graphing Polar
Equations in Matlab). Readers should have experience with parametric equations
and the arc length formula.
Polar Equations of Conic Sections
Abstract-
This activity is an interactive study of the polar form of the equation
for a conic section. Readers should be familiar with polar coordinates
and triangle trigonometry. This activity is also a vehicle for the introduction
of the Geometer's Sketchpad, though no prior experience with the
Sketchpad
is assumed.
Prerequisites-
-
Polar coordinates and right triangle trigonometry.
Lines and Planes in Matlab
Abstract-
In this activity you will learn how to plot lines and and planes.
Prerequisites-
-
Some knowledge of how to enter vectors and matrices in Matlab. Some familiarity
with Matlab’s array operations and plot command is useful.
Surfaces in Matlab
Abstract-
Matlab's mesh command is used to draw surfaces represented by equations
of the form z = f(x,y).
Prerequisites-
-
Some familiarity with Matlab's element wise operators (.*, .^, ./)
is required.
Cylindrical Coordinates and Quadric Surfaces
Abstract-
Cylindrical coordinates are used to graph quadric surfaces..
Prerequisites-
-
Familiarity with Matlab’s mesh command and Matlab’s array operations.
Curvature in Matlab
Abstract-
A discussion of curvature. Matlab is used to simplify computations and
depict visualizations that aid in understanding the concept of curvature.
Prerequisites-
-
Familiarity with Matlab’s mesh command and Matlab’s array operations.
Velocity and Acceleration---Constant Speed
Abstract-
If a particle moves along a path with constant speed, then its acceleration
and velocity vectors will be orthogonal at each position along the path.
Matlab is used to provide a clear visualization of this fact.
Prerequisites-
-
You must understand how to find the position, velocity, and acceleration
of a particle as it travels through space. Some familiarity with plotting
in Matlab. You'll also need a rudimentary understanding of Matlab's element-wise
operators (.*, .^, ./).
Level Curves in Matlab
Abstract-
Matlab is used to explore the level curve concept of functions mapping
R2 into R.
Prerequisites-
-
Some familiarity with Matlab's meshgrid command is required. Rudimentary
understanding of Matlab's element-wise operators (.*, .^, ./) is
required.
Linear Approximations
Abstract-
Linear approximations of functions g:R2-->R are explored through
the visualization powers of Matlab. A review of Taylor's theorem and its
application to multivariable functions is employed to find linear approximations.
Prerequisites-
-
Taylor's theorem for functions g:R-->R. Partial derivatives. Some familiarity
with Matlab's element wise operators (.*, ./, .^).
The Gradient
Abstract-
Matlab is used to explore properties of the gradient.
Prerequisites-
-
A basic understanding of the level curve concept. It is recommended that
you practice drawing contours in Matlab before attempting this project.
You should also know how to use Matlab's meshgrid and mesh
commands. Knowledge of Matlab's element wise operators (.*, ./, .^)
is required.
Max-Min and Saddle Points
Abstract-
In this activity you will use the Matlab software to find maximum, minimum,
and saddle points of a function.
Prerequisites-
-
Partial differentiation. Familiarity with Matlab's meshgrid,
mesh,
and contour commands. Familiarity with Matlab's element wise operators
(.*,
./, .^) is required. The Symbolic Toolbox is used as an aid in finding
critical points, but readers can complete the activity without it.