Questions: A function that satisfies Laplace's equation is called a harmonic function. (a) Is (f(x, y, z)=x^2+y^2-2 z^2) harmonic? What about (f(x, y, z)=x^2-y^2+z^2) ? (b) We may generalize Laplace's equation to functions of (n) variables as [ partial^2 f / partial x1^2 + partial^2 f / partial x2^2 + cdots + partial^2 f / partial xn^2 = 0 ] Give an example of a harmonic function of (n) variables, and verify that your example is correct.

A function that satisfies Laplace's equation is called a harmonic function.

(a) Is (f(x, y, z)=x^2+y^2-2 z^2) harmonic? What about (f(x, y, z)=x^2-y^2+z^2) ?

(b) We may generalize Laplace's equation to functions of (n) variables as

[
partial^2 f / partial x1^2 + partial^2 f / partial x2^2 + cdots + partial^2 f / partial xn^2 = 0
]

Give an example of a harmonic function of (n) variables, and verify that your example is correct.

Transcript text: that satisfies Laplace's equation is called a harmonic function. ${ }^{3}$ (a) Is $f(x, y, z)=x^{2}+y^{2}-2 z^{2}$ harmonic? What about $f(x, y, z)=x^{2}-y^{2}+z^{2}$ ? (b) We may generalize Laplace's equation to functions of $n$ variables as \[ \frac{\partial^{2} f}{\partial x_{1}^{2}}+\frac{\partial^{2} f}{\partial x_{2}^{2}}+\cdots+\frac{\partial^{2} f}{\partial x_{n}^{2}}=0 \] Give an example of a harmonic function of $n$ variables, and verify that your example is correct.

Solution

Solution Steps

Solution Approach

(a) To determine if a function is harmonic, we need to check if it satisfies Laplace's equation. For a function $ f(x, y, z) $, this means calculating the second partial derivatives with respect to each variable and summing them. If the sum is zero, the function is harmonic.

(b) For a function of $ n $ variables to be harmonic, it must satisfy the generalized Laplace's equation. We can construct a simple example, such as a function that is the sum of squares of the variables with appropriate coefficients, and verify it by checking if the sum of its second partial derivatives is zero.

Step 1: Check if $ f(x, y, z) = x^2 + y^2 - 2z^2 $ is Harmonic

To determine if the function $ f_1(x, y, z) = x^2 + y^2 - 2z^2 $ is harmonic, we compute the Laplacian:

\[ \Delta f_1 = \frac{\partial^2 f_1}{\partial x^2} + \frac{\partial^2 f_1}{\partial y^2} + \frac{\partial^2 f_1}{\partial z^2} = 0 \]

Since $ \Delta f_1 = 0 $, the function $ f_1 $ is harmonic.

Step 2: Check if $ f(x, y, z) = x^2 - y^2 + z^2 $ is Harmonic

Next, we check the function $ f_2(x, y, z) = x^2 - y^2 + z^2 $:

\[ \Delta f_2 = \frac{\partial^2 f_2}{\partial x^2} + \frac{\partial^2 f_2}{\partial y^2} + \frac{\partial^2 f_2}{\partial z^2} = 2 \]

Since $ \Delta f_2 = 2 \neq 0 $, the function $ f_2 $ is not harmonic.

Step 3: Example of a Harmonic Function of $ n $ Variables

We consider the function $ f(x_1, x_2, x_3, x_4) = x_1^2 - x_2^2 + x_3^2 - x_4^2 $ as an example of a harmonic function in four variables. We compute its Laplacian:

\[ \Delta f = \frac{\partial^2 f}{\partial x_1^2} + \frac{\partial^2 f}{\partial x_2^2} + \frac{\partial^2 f}{\partial x_3^2} + \frac{\partial^2 f}{\partial x_4^2} = 0 \]

Since $ \Delta f = 0 $, this function is indeed harmonic.

Final Answer

$ f_1(x, y, z) = x^2 + y^2 - 2z^2 $ is harmonic: $\boxed{\text{Yes}}$
$ f_2(x, y, z) = x^2 - y^2 + z^2 $ is harmonic: $\boxed{\text{No}}$
Example of a harmonic function of $ n $ variables: $ f(x_1, x_2, x_3, x_4) = x_1^2 - x_2^2 + x_3^2 - x_4^2 $: $\boxed{\text{Yes}}$

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