Multi-Variable Calculus Assignment

due Wednesday, February 21, 2001

Reading

Read in the text to make sure that you understand the form of the multi-variable chain rule for differentiation which says that

φ′(t) = (∇f)(R(t)) · R′(t) when \{ \begin{matrix}R is a point-valued function of t . \\ f is a scalar-valued function of points. \\ ∇f is the vector of partial derivatives of f . \\ φ = f ∘ R, i.e., φ(t) = f(R(t)) . \end{matrix}

Exercises

1. For n = 2 give a geometric interpretation of what the multi-variable chain rule means in the case of the example

R(t) = (cos t, sin t) , f(x, y) = x^{2} + y^{2}

by working through the following steps:

(a) Compute R′(t).

(b) Compute (∇f)(x, y) at a general point (x, y).

(c) Compute φ(t) = f(R(t)).

(d) Compute φ′(t).

(e) In what direction does the vector R′(t) point?

(f) In what direction does the vector (∇f)(R(t)) point?

(g) How are these two directions related?

(h) What fact in plane Euclidean geometry is illustrated here?

2. Let S be the parameterized surface that is defined by

(s, t) ⟼ (st, s^{2} − t^{2}, s^{2} + t^{2}) .

(a) Find parameter values s, t that yield the point (1, 0, 2).

(b) Find a vector that is normal to S at the point (1, 0, 2).

(c) Find the equation of the tangent to S at the point (1, 0, 2).

(d) Find an equation in x, y, z that is satisfied for all s, t when

x = st , y = s^{2} − t^{2} , z = s^{2} + t^{2} .

[Processed from GELLMU to L^{A}TEX on Fri 16 Feb 2001 at 08:28:39 EST]