u(x, y) is said to be harmonic if it is a solution to the partial differential equation ux x+uy

U(x, y) is said to be harmonic if it is a solution to the...

Key Concepts

Harmonic Functions

Harmonic functions are twice continuously differentiable functions that satisfy Laplace's equation. They have important applications in physics and engineering, particularly in modeling steady-state heat distribution and potential fields. The defining property of a harmonic function is that the sum of its second partial derivatives with respect to each independent variable is zero.

Laplace's Equation

Laplace's equation is a second-order partial differential equation expressed as u_xx + u_yy = 0 in two dimensions. It is used to identify harmonic functions and appears in various scientific fields, serving as a mathematical model for systems in equilibrium.

Partial Differentiation

Partial differentiation involves taking derivatives of functions with respect to one variable while keeping other variables constant. This process is essential for calculating the second partial derivatives required to verify that a function satisfies Laplace's equation.

Chain Rule

The chain rule is a method for calculating the derivative of a composite function. In the context of verifying harmonicity, it is used to differentiate expressions where functions are composed with other functions, such as in the logarithm of a sum of squares.

Second Partial Derivatives

Second partial derivatives are the results of differentiating a function twice with respect to one of its variables. In assessing whether a function is harmonic, the second derivatives with respect to each variable are calculated and summed to check if the total equals zero, as mandated by Laplace's equation.

Transcript

00:01 So if we want to show natural log of x squared plus y squared is harmonic, what we need to do is find the second partials with respect to x and y add it together and show this is going to be equal to zero.

00:15 So let's go ahead and find these partials.

00:17 So let's do with respect to x first.

00:21 So to take the derail of that, we'll just use chain rule.

00:24 So it'll be 1 over x squared plus y squared and then del by del x of x squared plus y squared.

00:32 And now remember we're going to assume y is a constant with respect to x.

00:36 So the derivative that is 0, the derivative this is 2x.

00:41 And so that would give us 2x over x squared plus y squared.

00:47 Now, if we want to take the derivative of this here, we're going to have a function that depends on x over a function that depends on x, which would mean we would want to use quotient rule.

00:59 So the second partial with respect to x is going to be.

01:03 So you have x squared plus y squared del by del x of 2x minus.

01:15 Then we switch them.

01:17 So 2x del by del x of x squared plus y squared all over what we have in the denominator squared.

01:30 So derivative of 2x is going to be not zero.

01:35 And then the derivative of x squared it's going to be 2x and then y squared well since y is a constant this is zero and now let's distribute the two and multiply those together so we get 2 x squared plus 2 y squared minus 4 x squared all over x squared plus y squared squared squared and now we can go ahead and combine those terms in the numerator so that gives us 2y squared minus 2 x squared all over x squared plus y squared squared squared squared.

02:19 All right, so now let's go ahead and find the second partial with respect to y...

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