Vector HW 7

Problem 1.  Verify that the Chain Rule (as stated in class) is true for the functions

\displaystyle F(x,y) = (x+y, ye^x) \text{ and } G(s, t) = (s+t, s-t, s^2).

Do this by first writing out their composition as a function H and then computing the derivative matrix DH.

Next, compute the derivative matrices DF and DG and multiply them.  Your final answer should equal DH.

Problem 2.  Consider the function f(x,y) = y/x.  Use the Chain Rule to compute \partial f/ \partial \theta where \theta is our standard cylindrical coordinate variable.

Problem 3.  (a) Suppose G: \mathbb{R}^3 \to \mathbb{R}^3 is a differentiable function and that at the point \vec{x}_0 = \left(1, 2, 1\right) we know that G(\vec{x}_0) = \left(0, 0, 2\right) and we know that

\displaystyle DG\big{|}_{\vec{x}_0} = \left[ \begin{array}{ccc} 1 & 2 & \pi \\ 0 & 1 & -1 \\ -1 & 7 & 1/2 \end{array} \right]

Compute the linear approximation, L(x,y,z), for G at the point \vec{x}_0.

(b) Suppose F:\mathbb{R}^3 \to \mathbb{R}^2 is a differentiable function and that at the point \vec{z}_0 = \left(0, 0, 2\right) we know that

\displaystyle DF \big{|}_{\vec{z}_0} = \left[ \begin{array}{ccc} 1 & 1 & 1 \\ \sqrt{2} & -2 & 5 \end{array}\right].

Compute the derivative matrix D\left(F\circ G\right) at the point \vec{x}_0 = (1, 2, 1) where G is the function from part (a).

Problem 4. Find the equation of the plane that passes through the point \vec{x}_0 = (1, 2, 4) and is parallel to the plane x - 4y + 2 = 0.

Problem 5.  Draw a picture of the set S = \left\{(x,y,z) \in \mathbb{R}^3 : x^2+4x + y^2 + z^2 = -3 \right\}.

Problem 6.  Draw a picture of the line parameterized by

\displaystyle \ell(t) = (1+t, 1+t).

Problem 7.  Consider the function f(x,y) = x^3(x^2-y^2).

(a) Compute the derivative matrix Df.

(b) Compute the gradient \nabla f.

(c) Form the new function g:\mathbb{R}\to\mathbb{R} defined by

\displaystyle g(t) = \left( f\circ \ell\right)\,(t) = f\left(\ell(t)\right)

where \ell(t) is the function that parameterized the line in problem 6.  Note that this function g(t) is the kind of function we might see in a Calculus I class; for old time’s sake, compute the derivative g'(t).

(d) Use the Chain Rule to compute the 1\times 1 derivative matrix

\displaystyle Dg

where g(t) is the same composed function from part (c).

Note:  Your computation for part (d) could have been re-phrased in terms of the gradient of f instead of its derivative matrix, Df.  Had we done this, we would have found

\displaystyle g'(t) = \frac{d}{dt}\, \left(f\circ \ell\right) = \nabla f \cdot \ell'(t).

This is an important note!  It is our definition of something called a directional derivative!  That is, we think of this computation as telling us how f changes at the point \ell(0) = (1, 1) when we move inputs in the direction of \ell!

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