Directional Derivative 2D  Contents

The directional derivative ∇Uf(x0,y0,z0) of the function f(x,y,z) at the point (x0,y0,z0) in the direction of the unit vector U=(cos(θ)cos(φ),sin(θ)cos(φ),sin(φ)) is the derivative of z(t)=(x0+t*cos(θ)cos(φ),y0+t*sin(θ)cos(φ), z0+t*sin(φ)) at t=0, i.e. ∇Uf(x0,y0,z0)=U⋅∇f(x0,y0,z0).

Figure1

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Exercises

  • 1. Evaluate the directional derivatives for each of the sets of conditions below:
    • f(x, y, z) = x + y + z, (x0, y0, z0) = (1, 0, -1), θ = 0, φ = 0
    • f(x, y, z) = x2 + y2 + z2, (x0, y0, z0) = (0, 0, 0), θ = π/4, φ = π/4
    • f(x, y, z) = x2 + sin(y) + ez, (x0, y0, z0) = (5, π/4, 1), θ = π/2, φ = 5π/6
  • 2. What restrictions must be set on the function f(x, y, z) in order for the directional derivative at the origin to be the same in all directions (hint: one of them is f(x, y, z) = f(-x, y, z))?


    • Second Partial Derivative 1D   2D  Top of Page  Contents

    A second partial derivative is a partial derivative of a function which is itself a partial derivative of another function.

    There are nine types of second partial derivatives for functions of three variables.

    Third, fourth, fifth, and in general, nth partial derivatives for any positive integer n, exist as well.

    Figure2

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    Exercises

  • 1. Find the following second partial derivatives as functions of x, y, and z:
    • fxz(x, y, z), where f(x, y, z) = x2 + xy + xz
    • fyy(x, y, z), where f(x, y, z) = x2yz + xy2z + xyz2
    • fzy(x, y, z), where f(x, y, z) = zey
    • fxy(x, y, z), where f(x, y, z) = 1/(xyz)
  • 2. Suppose the demo above did not provide graphs of the first and second partial derivatives. How could you use the graph of f(x, y, z) to determine what the graph of fxy(x, y, z) would look like?
  • 3. Consider the function f(x, y, z) = Axy + Bxz + Cyz. Describe the relationships between the mixed second partial derivatives. Will these relationships exist for any function of three variables f(x, y, z) or are there exceptions?


    • Taylor Series 1D   2D  Top of Page  Contents

    Taylor series are polynomials that approximate functions.

    For a function f of three variables, the Taylor series of f at a point depends on the partial derivatives at (x0, y0, z0).
    Figure4

    [D]

    Exercises

  • 1. Try entering for f(x, y, z) various polynomial functions of degree ≤ 2. What do you notice? Why should this result be expected?
  • 2. Enter for f(x, y, z) the expresssion x2 + y2 + z3. Describe the accuracy of each of the three slices of the Taylor approximation.


    • Constrained Maxima & Minima 1D   2D  Top of Page  Contents

    Given a contour g(x,y,z) = c of a function g, we can find the maximum value of the function f(x,y,z) on this contour by locating the points at which a contour of f is tangent to the contour of g.

    At such a point the gradient of f is parallel to the gradient of g, such that

    ∇f(x,y,z) = λ∇g(x,y,z)

    where λ is constant.

    figure13

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    figure12

    [D]



    Exercises

  • 1. Find the maximum and minimum values of x2 + y2 + z2 on the surface (x - 1/2)2 + y2 = 1/4.
  • 2. Find the maximum and minimum values of cos(x) + cos(y) - z on the surface z = 2.
  • 3. Find the maximum and minimum values of cos(x) + cos(y) - z on the surface x2+y2+z2 = 1.