22.8. THE GRADIENT 481

10. Suppose r1 (t) = (cos t,sin t, t) ,r2 (t) = (t,2t,1), and r3 (t) = (1, t,1). Find the rateof change with respect to t of the volume of the parallelepiped determined by thesethree vectors when t = 1.

11. A trash compactor is compacting a rectangular block of trash. The width is changingat the rate of −1 inches per second, the length is changing at the rate of −2 inchesper second and the height is changing at the rate of −3 inches per second. How fastis the volume changing when the length is 20, the height is 10, and the width is 10?

12. A trash compactor is compacting a rectangular block of trash. The width is changingat the rate of −2 inches per second, the length is changing at the rate of −1 inchesper second and the height is changing at the rate of −4 inches per second. How fastis the surface area changing when the length is 20, the height is 10, and the width is10?

13. The ideal gas law is PV = kT where k is a constant which depends on the number ofmoles and on the gas being considered. If V is changing at the rate of 2 cubic cm.per second and T is changing at the rate of 3 degrees Kelvin per second, how fast isthe pressure changing when T = 300 and V equals 400 cubic cm.?

14. Let S denote a level surface of the form f (x1,x2,x3) = C. Show that any smoothcurve in the level surface is perpendicular to the gradient.

15. Suppose f is a C1 function which maps U , an open subset of Rn one to one and ontoV , an open set in Rm such that the inverse map, f−1 is also C1. What must be true ofm and n? Why? Hint: Consider Example 22.6.5 on Page 479. Also you can use thefact that if A is an m×n matrix which maps Rn onto Rm, then m ≤ n.

16. Finish Example 22.6.4 by finding fy in terms of θ ,r. Show that fy = sin(θ) fr +cos(θ)

r fθ .

17. ∗Think of ∂x as a differential operator which takes functions and differentiates themwith respect to x. Thus ∂x f ≡ fx. In the context of Example 22.6.4, which is on polarcoordinates, and Problem 16, explain how

∂x = cos(θ)∂r −sin(θ)

r∂θ

∂y = sin(θ)∂r +cos(θ)

r∂θ

The Laplacian of a function u is defined as ∆u = uxx+uyy. Use the above observationto give a formula ∆u in terms of r and θ . You should get urr +

1r ur +

1r2 uθθ . This is

the formula for the Laplacian in polar coordinates.

22.8 The GradientHere we review the concept of the gradient and the directional derivative and prove theformula for the directional derivative discussed earlier.

Let f : U → R where U is an open subset of Rn and suppose f is differentiable on U .Thus if x ∈U ,

f (x+v) = f (x)+n

∑j=1

∂ f (x)∂xi

vi +o(v) . (22.15)