Mathematical Tools for Physics

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.