Let f(z)=u+i v be an analytic function, and let 𝐅 be the vector 𝐅=v 𝐢+u 𝐣. Show that the equatio

Let f(z)=u+i v be an analytic function, and let 𝐅 be the...

Key Concepts

Curl

Curl is a vector operator that measures the rotation or ‘twisting’ of a vector field around a point. A zero curl indicates that the field is irrotational, meaning it has no tendency to circulate about any point. In the analysis of analytic functions, equating curl to zero leads directly to relationships among the partial derivatives of the function’s components, which are equivalent to the Cauchy-Riemann equations.

Divergence

Divergence is a scalar measure in vector calculus that quantifies the magnitude of a source or sink at a given point in a vector field. It represents the net rate of flux expansion out of an infinitesimal volume and, when set to zero, implies that there is no net 'loss' or 'gain' of flow, which in the context of analytic functions reflects a balance tied to the Cauchy-Riemann conditions.

Cauchy-Riemann Equations

The Cauchy-Riemann equations are a set of two partial differential equations that provide necessary and sufficient conditions for a complex function to be analytic. They connect the partial derivatives of the real part and the imaginary part of the function, ensuring that the function’s complex derivative is independent of the direction of approach in the complex plane.

Analytic Function

An analytic function is a complex function that is differentiable at every point in its domain. This differentiability implies that the function can be locally represented by a convergent power series, and its real and imaginary parts possess continuous partial derivatives that satisfy additional interdependent conditions.

Transcript

00:01 Alrighty, let's first write down what the cauchy -riemann equations are.

00:05 If f of z is equal to u plus iv, where u and v are functions in x and y for real value at x and y, then the cauchy -riemann equations, let's call them cauchy -riemann equations, tell us that f is complex differentiable if and only if du dx, the partial derivative of u with respect to x, is equal to the partial derivative of v with respect to y.

01:05 So that has to be true, but what also has to be true is that partial derivative of u with respect to y has to be equal to the negative partial derivative of v with respect to x.

01:17 So when these two partial differential equations are satisfied, then you know that f is complex differentiable, and the other way around.

01:28 So now let me remind you what curl and divergence are.

01:32 Do this in red.

01:33 Curl is gradient vector cross some vector field.

01:45 In this case, f is given by vi plus uj, and divergence is gradient dot the vector field f.

02:05 So let's take care of curl first.

02:11 So i'm going to scroll down a little bit.

02:16 So we want curl of f of our field that we're given to equal zero, and that means we want the gradient cross f to equal zero.

02:36 And to do the cross product, we take now a little nuance here.

02:45 We're going to take the cross product as if f is three -dimensional.

02:49 Even though f is two -dimensional, we're going to say that f, which is equal to the vector vj, that's going to be congruent to v, excuse me, i said vj.

03:05 I meant to say vu.

03:07 I meant to say vu.

03:08 Sorry, let me start again.

03:10 All right.

03:11 So f is the vector vu, and that's congruent to vu zero.

03:24 So to compute the curl, we need a three -dimensional vector, and since we're only given a two -dimensional vector field, we're going to fix that by just appending a zero onto it in the third dimension, because you'll see we're going to use the scalar quantity that we get out of this curl.

03:41 That's what we want anyway, so it's valid to use this method.

03:46 And so now to compute it, we're going to take i -hat, j -hat, k -hat.

03:53 Then gradient is partial with respect to x, partial with respect to y, partial with respect to z.

04:04 And then f is given by v, u, and zero.

04:11 So we're going to compute this determinant, and it's supposed to be equal to zero.

04:17 All right...

Please give Ace some feedback