1. If it's given that A: R2 -> R is defined by A(x, y)={1, if xy greater than or equal to 0.
and 0, if xy < 0.
(a) how do i show that the partial derivatives Ax(0) and Ay(0) exist? (Note that Ax and Ay doesn't mean A*x and A*y. I just mean the first order derivatives.)
(b) How do I prove that A is not continuous at 0?
2. If A(x, y, z) = xzsin2y + ye^z
(a) How do I find all the mixed partial derivatives of A and verify Clairant's Theorem holds?
(b) Will Ayxz = Ayzx? Will I know this without actually calculating?
(c) Will Axyy=Ayzx?
(d) Will Ayzy=Azyy?
I'm having a bit of difficulty reading question 1.
Are you asking to show that
and exist?
For 1(b), "not continuous at 0"? I assume you mean the origin, the point (0,0)? There is no just "0" in .
To show is not continuous at , you need to show
. (In particular, this is true if the limit happens to not even exist, for example.)
So let's show . Recall that, with functions of one variable, there are only two ways to approach an -value: from the left or right. In , however, you can approach in infinitely many different ways; for example, along the -axis, the -axis, the parabola , etc, etc.
In this case, let's approach along the line (note that the point is actually on this line, otherwise this would make no sense). So we are computing along the line ; that is, the only points we are concerning ourselves with are those for which . Thus the limit becomes
.
But what is . We can assume because, on the line we are on, that would put us at the origin. But we can't be AT the origin if we are trying to take the limit as we APPROACH the origin. You can show (it's very, very easy...) that, whenever . This means . Therefore
.
Now, a limit exists if and only if it is the same along any path whatsoever. Since the limit along this path is 0, if the limit were to actually exist, it would have to be equal to 0. But , which means we would have
.
So the function cannot be continuous at the origin.
(NOTE: I'm not saying the limit IS 0; just that, by our above computation, it would have to be 0 IF IT EXISTED. You can actually show that the limit for this function does not exist at all.)
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Isn't question 2 straightforward? It's just asking you to compute partial derivatives.