Math 455
Homework 4
1. Change coordinates for the point or vector (3,5) from standard basis to the basis {(1,1), (2,4)}. Draw a graph
to compare the given point (vector) in both bases.
2. Change coordinates for the point or vector (2,5,-1) from the standard basis to the basis {(1,1,0), (1,0,1),
(0,1,1). Draw a graph (to the best of your ability) to compare the given point (vector) in both bases.
3. Physicists use “smooth” coordinate changes to simplify mathematics. In other words, the type of coordinates
should match the physical phenomenon. Here, “smooth coordinate change” means once differentiable.
Coordinate changes MUST leave calculated quantities the same (invariant) regardless of chosen coordinates.
a) Calculate the area under a constant function f(x) = 8 using both rectangular and polar coordinates.
Compare and contrast the methods.
x  r cos 
b) Given the polar coordinate change:
. Calculate the area under a semicircle of radius 5 in
y  r sin 
rectangular and polar coordinates. Compare and contrast the methods.
4. Use the fact that
x  r cos 
to find the inverse coordinate change. That is, find r and  both as functions of
y  r sin 
x and y.
5. Cylindrical coordinates in three dimensions are a direct generalization of polar coordinates in two
dimensions. See the picture below. Write out the cylindrical coordinates for the point (vector) (1,1,1) which is
assumed to be in rectangular coordinates. Hint: Problem 4.
Math 455
Homework 4
6. Spherical coordinates are calculated as below.
a) Write the equation of a sphere of radius 5 in both rectangular and spherical coordinates.
b) Change the point (vector) (1,1,1) in rectangular coordinates to spherical coordinates. Hint: First find
.
1
7. Suppose f ( x)   ( x  1) 2 dx . Use change of variables for the following, showing each step. Graph the
0
integrand before and after the change of variables, in each case to demonstrate how the area under the curve is
left invariant under coordinate change.