Problem 16 - The Calculus with Analytic Geometry

Question

In Exercises 15 through 18, show that \(u(x, y)\) satisfies the equation $$ \frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0 $$ which is known as Laplace's equation in \(R^{2}\). $$ u(x, y)=e^{x} \sin y+e^{y} \cos x $$

Step-by-Step Solution

Answer

\(u(x, y)\) satisfies Laplace's equation because \(\frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}} = 0\).

Step 1: Compute the first partial derivatives

First, find the partial derivatives of the function \(u(x, y) = e^{x} \sin y + e^{y} \cos x\).\ \frac{\partial u}{\partial x} = \frac{\partial}{\partial x} (e^{x} \sin y + e^{y} \cos x) = e^{x} \sin y - e^{y} \sin x.

Step 2: Compute the second partial derivative with respect to x

Next, find the second partial derivative of \(u\) with respect to \(x\).\ \frac{\partial^{2} u}{\partial x^{2}} = \frac{\partial}{\partial x} \left( e^{x} \sin y - e^{y} \sin x \right) = e^{x} \sin y - e^{y} \cos x.

Step 3: Compute the first partial derivatives

Now, find the partial derivatives with respect to \(y\).\ \frac{\partial u}{\partial y} = \frac{\partial}{\partial y} (e^{x} \sin y + e^{y} \cos x) = e^{x} \cos y + e^{y} (-\sin x).

Step 4: Compute the second partial derivative with respect to y

Finally, find the second partial derivatives of \(u\) with respect to \(y\).\ \frac{\partial^{2} u}{\partial y^{2}} = \frac{\partial}{\partial y} \left( e^{x} \cos y - e^{y} \sin x \right) = -e^{x} \sin y - e^{y} \cos x.

Step 5: Add the second partial derivatives

Add the second partial derivatives:\ \frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}} = (e^{x} \sin y - e^{y} \cos x) + (-e^{x} \sin y - e^{y} \cos x).\ This simplifies to: (e^{x} \sin y - e^{x} \sin y) + (-e^{y} \cos x - e^{y} \cos x) = 0.\ Thus, the function \(u(x, y)\) satisfies Laplace's equation.

Key Concepts

Partial Derivatives

Partial derivatives are used to measure how a function changes as only one of its input variables changes, holding the others constant. They are an extension of ordinary derivatives to functions of multiple variables. For example, if we have a function of two variables, like our problem's function, partial derivatives will assess how the function varies with respect to each variable independently. In the given exercise, we have a function \(u(x, y) = e^x \sin(y) + e^y \cos(x)\). We need to find partial derivatives by differentiating this function with respect to \(x\) and \(y\) separately.

Second Partial Derivatives

Second partial derivatives further extend the concept by differentiating a partial derivative. This helps reveal how the rate of change itself is changing. In the exercise, after finding the first partial derivatives of our function \(u(x, y)\), we proceed to compute the second partial derivatives. For example, the second partial derivative of \(u\) with respect to \(x\) is computed by differentiating \( \frac{\partial u}{\partial x}\) with respect to \(x\) again. Similarly, we find the second partial derivative with respect to \(y\). These second partial derivatives are essential in solving and confirming the solution to Laplace's equation.

Analytic Geometry

Analytic geometry, also known as coordinate geometry, uses algebraic methods to explore geometric problems. By representing geometric shapes with equations, it allows us to apply calculus and algebra to solve problems about curves, planes, and surfaces. In the context of the exercise, the function \(u(x, y)\) can be visualized geometrically in a 3-dimensional space, where we see how it behaves as a surface defined over the \(x\) and \(y\)-axes. Understanding the slope and curvature of this surface at each point involves computing partial and second partial derivatives.

Partial Differential Equations

Partial differential equations (PDEs) involve multiple independent variables and their partial derivatives. They are used in various fields like physics, engineering, and economics to describe how things change over space and time. Laplace's equation, \(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0\), is a classic PDE used to describe steady-state heat distribution, electric potential, and fluid flow. To solve this equation in our exercise, we demonstrated that the sum of the second partial derivatives with respect to both \(x\) and \(y\) equals zero, confirming that the given function satisfies Laplace's equation.