91Ó°ÊÓ

Partial derivatives are a fundamental concept in calculus, especially when dealing with functions of several variables. When we have a multivariable function, like in our exercise with variables \(x\), \(y\), and \(z\), each variable might influence the outcome independently. A partial derivative of a function is its derivative with respect to one of these variables, treating all the other variables as constant.

For the given function \(u = \frac{1}{\sqrt{x^2 + y^2 + z^2}}\), the partial derivative with respect to \(x\) is taken by differentiating \(u\) with \(x\) while keeping \(y\) and \(z\) constant, resulting in:

• \(u_x = -(x^2 + y^2 + z^2)^{-3/2} \cdot x\)

The same procedure applies for \(y\) and \(z\) to obtain \(u_y\) and \(u_z\).

Partial derivatives are crucial when checking if a function satisfies differential equations, such as the Laplace equation, by showing changes with respect to each variable separately.

Three-dimensional calculus extends calculus to three-dimensional space, involving functions with three variables. It's highly applicable in physics and engineering, where many processes occur in three dimensions. In our problem, we deal with a potential field represented by the function \(u = \frac{1}{\sqrt{x^2 + y^2 + z^2}}\), which is a common representation in electrostatics and fluid dynamics.

To explore functions in three dimensions, we use tools such as gradients, divergence, and Laplacians. The Laplace equation, \(u_{xx} + u_{yy} + u_{zz} = 0\), is especially important in finding scalar potentials where equilibrium conditions hold, meaning there's no change in potential in any direction. Thus, checking each second-order partial derivative implies exploring how the function changes shape or size when moving along the \(x\), \(y\), and \(z\) axes.

By confirming \(u_{xx} + u_{yy} + u_{zz} = 0\), we affirm that the function `u` represents a harmonic function, important when solving real-world physics problems involving three dimensions.

The chain rule is a powerful tool in calculus used for differentiating composite functions. It allows us to differentiate functions nested within other functions by following a simple principle: differentiate the outer function and then multiply by the derivative of the inner function.

In the given exercise, our function is expressed as \(u = (x^2 + y^2 + z^2)^{-1/2}\). To find \(u_x\), \(u_y\), and \(u_z\), the chain rule assists in handling the exponentiation and square root operations.

• First, differentiate \((x^2 + y^2 + z^2)^{-1/2}\) as if it were a simple power function, \(-1/2 (x^2 + y^2 + z^2)^{-3/2}\).
• Then multiply by the derivative of the inner function, \(2x\) for \(x\), \(2y\) for \(y\), and \(2z\) for \(z\).

The chain rule simplifies complex derivative operations, breaking them into easier steps, and it's a cornerstone of solving multivariable calculus problems efficiently.