8—Multivariable Calculus 205Exercises1 For the functionsf(x,y) =Axy^2 sin(xy), x(t) =Ct^3 , y(t) =Dt^2 , computedf/dttwo ways.
First use the chain rule, then do explicit substitution and compute it directly.
2 Compute
(
∂f/∂x
)
yand(
∂f/∂y
)
xfor(a)f(x,y) =x^2 − 2 xy+y^2 , (b)f(x,y) = ln(y/x), (c)f(x,y) = (y+x)/(y−x)
3 Computedf/dxusing the chain rule for
(a)f(x,y) = ln(y/x), y=x^2 , (b)f(x,y) = (y+x)/(y−x), y=αx,
(c)f(x,y) = sin(xy), y= 1/x
Also calculate the results by substitutingyexplicitly and then differentiating, comparing the results.
4 Letf(x,y) =x^2 − 2 xy, and the polar coordinates arex=rcosφ, y=rsinφ. Compute
(
∂f
∂x
)
y,
(
∂f
∂y
)
x,
(
∂f
∂x
)
r,
(
∂f
∂y
)
r,
(
∂f
∂x
)
φ,
(
∂f
∂y
)
φ5 Letf(x,y) =x^2 − 2 xy, and the polar coordinates arex=rcosφ, y=rsinφ. Compute
(
∂f
∂r
)
φ,
(
∂f
∂φ
)
r,
(
∂f
∂r
)
x,
(
∂f
∂φ
)
x,
(
∂f
∂r
)
y,
(
∂f
∂φ
)
y6 For the functionf(u,v) =u^3 −v^3 , what is the value at(u,v) = (2,1)? Approximately what is its
value at(u,v) = (2. 01 , 1 .01)? Approximately what is its value at(u,v) = (2. 01 , 0 .99)?
7 Assume the Earth’s atmosphere is uniform density and 10 km high, what is its volume? What is the
ratio of this volume to the Earth’s volume?
8 For a cube 1 m on a side, what volume of paint will you need in order to paint it to a thickness of
0.2 mm? Don’t forget to paint all the sides.
9 What isgradr^2? Do it in both rectangular and polar coordinates. Two dimensions will do. Are your
resultsreallythe same?
10 What isgrad
(
αx^2 +βy^2
)
. Do this in both rectangular and polar coordinates. For the polar form,
putxandyin terms ofrandφ, then refer to Eq. (8.27) for the polar form of the gradient. Finally,
compare the two results.
11 The Moon has a radius about 1740 km and its distance from Earth averages about 384 000 km from
Earth. What solid angle does the Moon subtend from Earth? What solid angle does Earth (radius
6400 km) subtend from the Moon?
12 Express the cylindrical unit vectorsˆr,φˆ,ˆzin terms of the rectangular ones. And vice versa.
13 Evaluate the volume of a sphere by integration in spherical coordinates.