8—Multivariable Calculus 241
Problems
8.1 Letr=
√
x^2 +y^2 ,x=Asinωt,y=Bcosωt. Use the chain rule to compute the derivative with respect
totofekr. Notice the various checks you can do on the result, verifying (or disproving) your result.
8.2 Sketch these functions* in plane polar coordinates:
(a)r=acosθ (b)r=asecθ (c)r=aθ (d)r=a/θ (e)r^2 =a^2 sin 2θ
8.3 The two coordinatesxandyare related byf(x,y) = 0. What is the derivative ofywith respect toxunder
these conditions? [What isdfalong this curve? And have you drawn a sketch?] Make up a test function (with
enough structure to be a test but still simple enough to verify your answer independently) and see if your answer
is correct. Ans:−(∂f/∂x)
/
(∂f/∂y)
8.4 Ifx=u+vandy=u−v, show that
(
∂y
∂x
)
u
=−
(
∂y
∂x
)
v
8.5 Ifx=rcosθandy=rsinθ, compute
(
∂x
∂r
)
θ
and
(
∂x
∂r
)
y
8.6 What is the differential off(x,y,z) = ln(xyz).
8.7 Iff(x,y) =x^3 +y^3 and you switch to plane polar coordinates, use the chain rule to evaluate
(
∂f
∂r
)
θ
,
(
∂f
∂θ
)
r
,
(
∂^2 f
∂r^2
)
θ
,
(
∂^2 f
∂θ^2
)
r
,
(
∂^2 f
∂r∂θ
)
,
Check one or more of these by substitutingrandθexplicitly and doing the derivatives.
* Seehttp://www-groups.dcs.st-and.ac.uk/~history/Curves/Curves.htmlfor more.