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$u(x, y)$ is said to be harmonic if it is a solution to the partial differential equation $u_{x x}+u_{y y}=0,$ where $c$ is a constant.Show the given function is harmonic.$$u(x, y)=x^{2}-y^{2}$$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 2

Partial Derivatives

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Lectures

12:15

In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

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$u(x, y)$ is said to be ha…

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A function of two variable…

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So if we want to show that this function X squared minus twice, where is harmonic? We just need to plug it in to this equation up top here. So the second partial derivative with respect to X plus the second partial derivative perspective. Why? And we want to show this is equal to zero. So let's go ahead and just first find our partial derivatives. So I'll do del by Dell experts, so find our process with respect to X. So use of X. So remember, we treat this Why here? As if it's a constant. So the derivative of y skirt, which is B zero and then the derivative of expert, is going to be two X And then if we take the partial of this with respect X again. So we get the second personal with respect to X is going to be, well, two x time or take the derivative. That would just be too. Uh huh. Now, if instead of doing the partial but respect X, we were to do Actually, I'll just go ahead and rewrite this Odell Beidle Why? I hope so. We have you X Y is equal to X squared minus y squared. So now we're going to assume expert is a constant. So when we take the derivative that that's just gonna be zero, that we do the derivative of y Swear to Why? So we end up with you. Supply is negative two y and then take the derivative of this with respect. Why? To get our second hole partial and the derivative of negative two y is just negative too. So now we add these together. So you xx plus you. Why? Why? So that is going to be two plus minus two, which is zero. So this checks the harmonic equation.

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