- Find f
_{x},f_{y},f_{z},f_{xy},f_{yx},f_{xz,}f_{zx},f_{zy},f_{yz}for the following. Verify the mixed partial derivatives are equal.- x
^{2}y^{3}z^{4}+ sin - sin+ x
^{2}yz - z ln
- e
^{x2+y2+z2 } - tan

- x
- Suppose f is a continuous function and f : U → ℝ where U is an open set and suppose that x ∈ U
has the property that for all y near x, f≤ f. Prove that if f has all of its partial derivatives at x, then f
_{xi}= 0 for each x_{i}. Hint: This is just a repeat of the usual one variable theorem seen in beginning calculus. You just do this one variable argument for each variable to get the conclusion. - As an important application of Problem 2 consider the following. Experiments are done at n times,
t
_{1},t_{2},,t_{n}and at each time there results a collection of numerical outcomes. Denote by_{i=1}^{p}the set of all such pairs and try to find numbers a and b such that the line x = at + b approximates these ordered pairs as well as possible in the sense that out of all choices of a and b, ∑_{i=1}^{p}^{2}is as small as possible. In other words, you want to minimize the function of two variables f≡∑_{i=1}^{p}^{2}. Find a formula for a and b in terms of the given ordered pairs. You will be finding the formula for the least squares regression line. - Show that if v= u, then v
_{x}= αu_{x}and v_{y}= βu_{y}. State and prove a generalization to any number of variables. - Let f be a function which has continuous derivatives. Show that u= fsolves the wave equation u
_{tt}− c^{2}Δu = 0. What about u= f? - D’Alembert found a formula for the solution to the wave equation u
_{tt}= c^{2}u_{xx}along with the initial conditions u= f,u_{t}= g. Here is how he did it. He looked for a solution of the form u= h+ kand then found h and k in terms of the given functions f and g. He ended up with something likeFill in the details.

- Determine which of the following functions satisfy Laplace’s equation.
- x
^{3}− 3xy^{2} - 3x
^{2}y − y^{3} - x
^{3}− 3xy^{2}+ 2x^{2}− 2y^{2} - 3x
^{2}y − y^{3}+ 4xy - 3x
^{2}− y^{3}+ 4xy - 3x
^{2}y − y^{3}+ 4y - x
^{3}− 3x^{2}y^{2}+ 2x^{2}− 2y^{2}

- x
- Show that z = is a solution to x+ y= z.
- Show that if Δu = λu where u is a function of only x, then e
^{λt}u solves the heat equation u_{t}− Δu = 0. - Show that if a,b are scalars and u,v are functions which satisfy Laplace’s equation then au + bv also satisfies Laplace’s equation. Verify a similar statement for the heat and wave equations.
- Show that u=e
^{−x2∕4c2t }solves the heat equation u_{t}= c^{2}u_{xx}.

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