Calculus of Multivariable Functions: Surfaces

Trivium

Sketch the contour plot and the graph of a multivariable function.

Prove that the limit of a multivariable function \( \lim_{(x,y)\to(a,b)} f(x,y) \) either exists or doesn’t exist, and if it exists compute its value.

Compute the partial derivatives of a multivariable function.

Given a surface either defined parametrically or defined as the graph \(z = f(x,y)\) of a multivariable function, determine an equation of the plane tangent to the surface at a point .

Exercises

1. Sketch the level surfaces for each of the following definitions of a function \(f\colon \mathbf{R}^2 \to \mathbf{R},\) and use those level surfaces to describe the graph of the function.

   \(\displaystyle f(x,y) = xy\)

   \(\displaystyle f(x,y) = \frac{1}{xy}\)

   \(\displaystyle f(x,y) = \frac{1}{x-y}\)

   \(\displaystyle f(x,y) = 6-3x-2y\)

   \(\displaystyle f(x,y) = y\mathrm{e}^x\)

   \(\displaystyle f(x,y) = \sqrt{9-x^2-y^2}\)

   \(\displaystyle f(x,y) = 4x^2+y^2+1\)

2. Describe the level surfaces for each of the following functions \(f\colon \mathbf{R}^3 \to \mathbf{R}.\)

   \(\displaystyle f(x,y,z) = x^2+y^2+z^2\)

   \(\displaystyle f(x,y,z) = x^2-y-z^2\)

3. Show that \(\lim\limits_{(x,y) \to (0,0)}\frac{x^2-y^2}{x^2+y^2}\) doesn’t exist.
4. Does \(\lim\limits_{(x,y) \to (0,0)}\frac{xy}{x^2+y^2}\) exist?
5. Does \(\lim\limits_{(x,y) \to (0,0)}\frac{xy^2}{x^2+y^4}\) exist?
6. Evaluate \(\lim\limits_{(x,y) \to (1,2)} x^2y^3-x^3y^2+3x+2y.\)
7. Evaluate \(\lim\limits_{(x,y) \to (-2,3)}\frac{x^2y+1}{x^3y^2-2x}.\)
8. Evaluate \(\lim\limits_{(x,y) \to (0,0)}\frac{3x^2y}{x^2+y^2}.\)
9. Evaluate the following limits:

   \(\displaystyle \lim\limits_{(x,y) \to (\pi,\pi/2)} y\sin(x-y)\)

   \(\displaystyle \lim\limits_{(x,y) \to (1,2)} \frac{2x-y}{4x^2-y^2}\)

   \(\displaystyle \lim\limits_{(x,y) \to (0,0)} \frac{5xy}{x^2+3y^2}\)

   \(\displaystyle \lim\limits_{(x,y) \to (TK)} TK\)

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. Determine a formula for the plane tangent to the elliptic paraboloid \(z = 2x^2+y^2\) at the point \((1,1,3).\)
18. Determine equations for (1) the tangent plane and (2) the normal line to the ellipsoid \(\frac{1}{4}x^2 + y^2 + \frac{1}{9}z^2 = 3\) at the point \((-2,1,-3).\)
19. What is a formula for the plane tangent to the surface defined parametrically as \((x,y,z) = \bigl(s^2, t^2, s+2t\bigr) \) at the point \((1,1,3)?\)

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 \[ \frac{\partial^2 u}{\partial \rho^2} + \frac{2}{\rho} \frac{\partial u}{\partial \rho} + \frac{\cot(\varphi)}{\rho^2} \frac{\partial u}{\partial \varphi} + \frac{1}{\rho^2} \frac{\partial^2 u}{\partial \varphi^2} + \frac{1}{\rho^2\sin^2(\varphi)} \frac{\partial^2 u}{\partial \theta^2} = 0 \,. \]
3. James Stewart Suppose \(f\) is a differentiable function of one variable. Show that all tangent planes to the surface \(z = xf(y/x)\) intersect in a common point.
4. James Stewart Among all planes that are tangent to the surface \(xy^2z^2 = 1,\) find the ones that are farthest from the origin.