Stuck in four variable problem

[Manolito]

Hi all:

I've a four-variable home assignment but in this occasion I'm a bit lost as to how to get the third and fourth partial derivatives. The problem is:

\(\displaystyle \mbox{Let }\, c\, \mbox{ be a non-zero constant and let }\, f:\mathbb{R}^4 \Rightarrow \mathbb{R}\, \)\(\displaystyle \mbox{ be the following:}\)

. . . . . . . .\(\displaystyle f(x,\, y,\, z,\, t)\, =\, \sin(x\, -\, ct)\, +\, \sin\left(\frac{3}{5}y\, +\, \frac{4}{5}z\, -\, ct\right)\)

\(\displaystyle \mbox{Show that }\displaystyle{\, f_{xx}\, +\, f_{yy}\, +\, f_{zz}\, -\, \left(\frac{1}{c^2}\right)f_{t\,t}\, =\, 0}\)

Any assistance would be very much appreciated.

Cheers
Manolo

 

[HallsofIvy]

Hi all:

I've a four-variable home assignment but in this occasion I'm a bit lost as to how to get the third and fourth partial derivatives.

??? The problem you post is does NOT ask for third and fourth derivatives.

The problem is:

\(\displaystyle \mbox{Let }\, c\, \mbox{ be a non-zero constant and let }\, f:\mathbb{R}^4 \Rightarrow \mathbb{R}\, \)\(\displaystyle \mbox{ be the following:}\)

. . . . . . . .\(\displaystyle f(x,\, y,\, z,\, t)\, =\, \sin(x\, -\, ct)\, +\, \sin\left(\frac{3}{5}y\, +\, \frac{4}{5}z\, -\, ct\right)\)

\(\displaystyle \mbox{Show that }\displaystyle{\, f_{xx}\, +\, f_{yy}\, +\, f_{zz}\, -\, \left(\frac{1}{c^2}\right)f_{t\,t}\, =\, 0}\)

Any assistance would be very much appreciated.

Cheers
Manolo

What have you done? Have you found the first and second derivatives with respect to these variables?

I'll start you off with the easy one. Since "x" appears only in the first sine, the first derivative of f with respect to x is the derivative of sine (which is cosine) times the derivative of x- ct with respect to x- which is 1. The first derivative of f, with respect to x, is cos(x- ct)(1)= cos(x- ct). The second derivative is the derivative of cosine, which is -sine, again times 1. The second derivative of f, with respect to x, is -cos(x- ct).