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

phi'(t) = (\nabla f)(R(t)) . R'(t) when
{
R is a point-valued function of t .
f is a scalar-valued function of points.
\nabla f is the vector of partial derivatives of f .
phi = f \circ R , i.e., phi(t) = f(R(t)) .

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:

    1. Compute R'(t).

    2. Compute (\nabla f)(x, y) at a general point (x, y).

    3. Compute phi(t) = f(R(t)).

    4. Compute phi'(t).

    5. In what direction does the vector R'(t) point?

    6. In what direction does the vector (\nabla f)(R(t)) point?

    7. How are these two directions related?

    8. 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}) .

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

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

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

    4. 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} .

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