- Find fx,fy,fz,fxy,fyx,fxz,fzx,fzy,fyz for the following. Verify the mixed partial derivatives
- x2y3z4 + sin
- sin +
- z ln
- 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 fxi = 0 for each
xi. 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
- As an important application of Problem 2 consider the following. Experiments are done at n times,
,tn and at each time there results a collection of numerical outcomes. Denote by
i=1p 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,
2 is as small as possible. In other words, you want to minimize the function of
two variables f
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 =
vx = αux and vy = βuy. State and prove a generalization to
any number of variables.
- Let f be a function which has continuous derivatives. Show that u =
f solves the wave
utt − c2Δu = 0. What about u =
- D’Alembert found a formula for the solution to the wave equation utt = c2uxx along with the initial
conditions u =
g. Here is how he did it. He looked for a solution of the form
k and then found
h and k in terms of the given functions f and g. He
ended up with something like
Fill in the details.
- Determine which of the following functions satisfy Laplace’s equation.
- x3 − 3xy2
- 3x2y − y3
- x3 − 3xy2 + 2x2 − 2y2
- 3x2y − y3 + 4xy
- 3x2 − y3 + 4xy
- 3x2y − y3 + 4y
- x3 − 3x2y2 + 2x2 − 2y2
- Show that z = is a solution to
- Show that if Δu = λu where u is a function of only x, then eλtu solves the heat equation
ut − 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
- Show that u =
solves the heat equation ut = c2uxx.