Calculus of Multivariable Functions: Surfaces

Trivium

Sketch the surface and contour plot of the graph \(z = f(x,y)\) of a multivariable function \(f\colon \mathbf{R}^2 \to \mathbf{R}.\)

Given a multivariable function \(f\colon \mathbf{R}^2 \to \mathbf{R}\) such that \[ \lim\limits_{(x,y)\to(a,b)} f(x,y) \] doesn’t exist, exhibit two paths in \(\mathbf{R}^2\) terminating at \((a,b)\) along which the outputs of \(f(x,y)\) approach different values.

Compute the partial derivatives of a multivariable function \(f\colon \mathbf{R}^2 \to \mathbf{R}.\)

Given an equation involving the symbols \(x,\) \(y,\) \(z,\) etc, be able to compute the partial derivative of any of them with respect to any other, both in the case that those symbols represent variables and in the case that they represent functions of other variables (e.g. \(t\) or \(s\)). If possible, be able to solve this implicit partial derivative to express the partial derivative explicitly.

Exercises

  1. For each of the following functions, sketch its graph and sketch a selection of its level curves.
    \(\displaystyle f(x,y) = 6-3x-2y\)
    \(\displaystyle g(x,y) = \sqrt{9-x^2-y^2}\)
    \(\displaystyle h(x,y) = 4x^2+y^2+1\)
  2. Describe the level surfaces of \(f(x,y,z) = x^2+y^2+z^2.\)
  3. Describe the level surfaces of \(f(x,y,z) = x^2-y-z^2.\)
  4. Show that \(\lim\limits_{(x,y) \to (0,0)}\frac{x^2-y^2}{x^2+y^2}\) doesn’t exist.
  5. Does \(\lim\limits_{(x,y) \to (0,0)}\frac{xy}{x^2+y^2}\) exist?
  6. Does \(\lim\limits_{(x,y) \to (0,0)}\frac{xy^2}{x^2+y^4}\) exist?
  7. Evaluate \(\lim\limits_{(x,y) \to (1,2)} x^2y^3-x^3y^2+3x+2y.\)
  8. Evaluate \(\lim\limits_{(x,y) \to (-2,3)}\frac{x^2y+1}{x^3y^2-2x}.\)
  9. Evaluate \(\lim\limits_{(x,y) \to (0,0)}\frac{3x^2y}{x^2+y^2}.\)
  10. For \(f(x,y) = x^3+x^2y^3-2y^2,\) calculate \(f_x(2,1)\) and \(f_y(2,1).\)
  11. For \(f(x,y) = \sin\Bigl(\frac{x}{1+y}\Bigr)\) calculate \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}.\)
  12. For \(f(x,y) = 4-x^2+-2y^2\) calculate \(f_x(1,1)\) and \(f_y(1,1)\) and interpret these numbers as slopes.
  13. Suppose \(z\) is defined implicitly as a function of \(x\) and \(y\) by the equation \(x^3+y^3+z^3+6xyz+4=0.\) Calculate \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\) and evaluate these partial derivatives at \((-1,1,2).\)
  14. For \(f(x,y,z) = \mathrm{e}^{xy}\ln(z),\) determine formulas for \(f_x\) and \(f_y\) and \(f_z.\)
  15. Determine formulas for all the second partial derivatives of \(f(x,y) = x^3+x^2y^3-2y^2.\)
  16. For \(f(x,y,z) = \sin(3x+yz),\) determine a formula for \(f_{xxyz}.\)
  17. If \(x = \sin(2t)\) and \(y = \cos(t)\) and \(z = x^2y+3xy^4,\) calculate the value of \(\frac{\mathrm{d}z}{\mathrm{d}t}\) when \(t = 0.\)
  18. If \(x = st^2\) and \(y = s^2t\) and \(z = \mathrm{e}^x\sin(y),\) determine formulas for \(\frac{\partial z}{\partial s}\) and \(\frac{\partial z}{\partial t}.\)
  19. If \(x = rs\mathrm{e}^t\) and \(y = rs^2\mathrm{e}^{-t}\) and \(z = r^2s\sin(t)\) and \(\omega = x^4y + y^2z^3,\) calculate the value of \(\frac{\partial \omega}{\partial s}\) when \(r = 2\) and \(s = 1\) and \(t = 0.\)
  20. If \(x^3+y^3 = 6xy,\) calculate a formula for \(\frac{\mathrm{d}y}{\mathrm{d}x}.\)

Problems & Challenges

  1. James Stewart For what values of the number \(r\) is the following function continuous on \(\mathbf{R}^3?\) \[ f(x,y,z) = \begin{cases} \frac{(x+y+z)^r}{x^2+y^2+z^2} \quad&\text{ if } (x,y,z) \neq 0 \\ 0 \quad&\text{ if } (x,y,z) = 0 \end{cases} \]

  2. James Stewart Consider Laplace’s equation \[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} = 0\,. \]

    1. Show that, when written in cylindrical coordinates, Laplace’s equation becomes \[ \frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{1}{r^2} \frac{\partial^2 u}{\partial \theta^2} + \frac{\partial^2 u}{\partial z^2} = 0 \,. \]
    2. Show that, when written in spherical coordinates, Laplace’s equation becomes \[ \frac{\partial^2 u}{\partial \rho^2} + \frac{2}{\rho} \frac{\partial u}{\partial \rho} + \frac{\cot(\varphi)}{\rho^2} \frac{\partial u}{\partial \varphi} + \frac{1}{\rho^2} \frac{\partial^2 u}{\partial \varphi^2} + \frac{1}{\rho^2\sin^2(\varphi)} \frac{\partial^2 u}{\partial \theta^2} = 0 \,. \]