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A function z= f(x,y) that obeys the partial differential equation z_{xx} + z_{yy}= 0 (known as...

Question:

A function z= f(x,y) that obeys the partial differential equation {eq}z_{xx} + z_{yy}= 0 {/eq} (known as Laplace's 2-dimensional equation) for all (x,y) in some open domain is said to be harmonic on that domain. Determine if {eq}z= e^x\sin(y) {/eq} is harmonic and state its domain.

Harmonic Functions:

Harmonic functions have applications in areas such as plate stress analysis, two-dimensional fluid flow and electrostatics.

In mathematical terms, it can be written in two variables such as: {eq}z = z\left( {x,y} \right) \to {z_{xx}} + {z_{yy}} = 0 {/eq}.

Answer and Explanation: 1

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First, we calculate the first partial derivative of the function:

{eq}z = {e^x}\sin y\\ \\ {z_x} = {e^x}\sin y\;,\;{z_y} = {e^x}\cos y {/eq}

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Higher-Order Partial Derivatives Definition & Examples

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Chapter 14 / Lesson 2
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In this lesson, we define the partial derivative and then extend this concept to find higher-order partial derivatives. Examples are used to expand your knowledge and skill.


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