Problem 43 - Thomas Calculus

Question

Green's Theorem and Laplace's equation Assuming that all the necessary derivatives exist and are continuous, show that if \(f(x, y)\) satisfies the Laplace equation $$\frac{\partial^{2} f}{\partial x^{2}}+\frac{\partial^{2} f}{\partial y^{2}}=0,$$ then $$\oint_{C} \frac{\partial f}{\partial y} d x-\frac{\partial f}{\partial x} d y=0$$ for all closed curves \(C\) to which Green's Theorem applies. (The converse is also true: If the line integral is always zero, then \(f\) satisfies the Laplace equation.)

Step-by-Step Solution

Answer

By Green's Theorem, the line integral is zero, confirming the statement.

Step 1: Understand Green's Theorem

Green's Theorem relates a closed line integral around a simple closed curve C to a double integral over the plane region D bounded by C. In the form we need, Green's Theorem states that: \( \oint_{C} P\,dx + Q\,dy = \iint_{D} \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right)\, dA \).

Step 2: Connect Line Integral with Green's Theorem

In this problem, consider \( P = \frac{\partial f}{\partial y} \) and \( Q = -\frac{\partial f}{\partial x} \) for the line integral \( \oint_{C} \frac{\partial f}{\partial y} \, dx - \frac{\partial f}{\partial x} \, dy \). According to Green's Theorem, this will equal \( \iint_{D} \left( \frac{\partial}{\partial x} \left(-\frac{\partial f}{\partial x}\right) - \frac{\partial}{\partial y} \left(\frac{\partial f}{\partial y}\right) \right)\, dA \).

Step 3: Simplify the Expression

The expression inside the double integral becomes \( \frac{\partial}{\partial x} \left(-\frac{\partial f}{\partial x}\right) - \frac{\partial}{\partial y} \left(\frac{\partial f}{\partial y}\right) = -\frac{\partial^{2} f}{\partial x^{2}} - \frac{\partial^{2} f}{\partial y^{2}} \).

Step 4: Apply Laplace's Equation

Since \( f(x, y) \) satisfies the Laplace's equation \( \frac{\partial^{2} f}{\partial x^{2}} + \frac{\partial^{2} f}{\partial y^{2}} = 0 \), substitute this into the calculated expression: \( -\frac{\partial^{2} f}{\partial x^{2}} - \frac{\partial^{2} f}{\partial y^{2}} = 0 \).

Step 5: Conclude Using Integrals

This implies the double integral \( \iint_{D} 0\, dA = 0 \). Therefore, according to Green's Theorem, the original closed line integral \( \oint_{C} \frac{\partial f}{\partial y} \, dx - \frac{\partial f}{\partial x} \, dy = 0 \) for any closed curve C to which Green's Theorem applies.

Key Concepts

Laplace's Equation

The Laplace's Equation is a second-order partial differential equation that is very important in various fields such as physics and engineering. This equation is expressed as:\[\frac{\partial^{2} f}{\partial x^{2}} + \frac{\partial^{2} f}{\partial y^{2}} = 0\]where \( f(x, y) \) is a function of two variables, and the partial derivatives \( \frac{\partial^{2} f}{\partial x^{2}} \) and \( \frac{\partial^{2} f}{\partial y^{2}} \) represent the curvature of \( f \) in the \( x \) and \( y \) directions, respectively.
Laplace's equation has a significant role: it models systems in equilibrium, where there is no net change occurring. This means the equation holds a balance between the rates of change in all involved directions. Because of this property, solutions to Laplace's equation are often called harmonic functions, and they are known to have smooth characteristics with no sudden changes. Understanding and solving this equation is crucial in applications ranging from electromagnetic fields to heat distribution.

Line Integral

A Line Integral is a way to integrate a function along a curve, taking into account both the path taken and the value of the function at different points along the path. For functions in two dimensions, a line integral can be expressed as:\[\oint_{C} P \, dx + Q \, dy\]where \( C \) is a closed curve or path, and \( P \) and \( Q \) are functions of \( x \) and \( y \).
In the context of vector fields, the line integral helps to calculate the work done by a force field in moving an object along a curve. In this exercise, the components of the vector field are \( \frac{\partial f}{\partial y} \) and \( -\frac{\partial f}{\partial x} \).
A significant feature of line integrals is their dependency on the path we choose to integrate over, not just the endpoints like traditional integrals. Green's Theorem heavily relies on line integrals to relate them to the double integral over the region enclosed by the curve.

Partial Derivatives

Partial Derivatives are crucial concepts in calculus used to measure how a function changes as only one of its inputs changes, keeping the other inputs constant. If we have a function \( f(x, y) \), its partial derivatives are denoted as:

• \( \frac{\partial f}{\partial x} \) for the change of the function with respect to \( x \) while keeping \( y \) constant.
• \( \frac{\partial f}{\partial y} \) for the change of the function with respect to \( y \) while keeping \( x \) constant.

Understanding partial derivatives is essential for functions of several variables, especially in describing physical phenomena where variables interact but may change independently.
This concept underpins Green's Theorem and Laplace's Equation, as both make use of partial derivatives to express changes and balance within a system. The second partial derivatives, as featured in Laplace's, provide information about the curvature or nature of the equilibrium within the respective system.