Derivation of Area Relation

Oct 13, 2025 by miodeya.com 65 views

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Introduction

In the realm of multivariable calculus, Green's Theorem plays a pivotal role in establishing a profound connection between line integrals and double integrals. This theorem, formulated by George Green in 1833, has far-reaching implications in various branches of mathematics and physics. In this article, we will delve into the application of Green's Theorem to derive a fundamental relation for the area enclosed by a closed curve in the plane.

Background and Notation

Before we embark on the derivation, let's establish some necessary notation and background. We are given a closed plane regular curve $\gamma(t)=\gamma(t,\tau) \in \mathbb{R}^2$ , where $t$ represents time and $\tau$ is a parameter with periodicity $T$ . The curve $\gamma(t)$ can be expressed in terms of its Cartesian coordinates as $\gamma(t) = (x(t), y(t))$ . Our objective is to find a relation for the area enclosed by this curve using Green's Theorem.

Green's Theorem

Green's Theorem states that for a simple closed curve $\gamma$ enclosing a region $D$ in the plane, the following equality holds:

\int_{\gamma} (Pdx + Qdy) = \iint_{D} \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dA

where $P$ and $Q$ are functions of $x$ and $y$ that are continuously differentiable in the region $D$ .

Derivation of Area Relation

To derive the relation for the area enclosed by the curve, we will utilize Green's Theorem with a specific choice of functions $P$ and $Q$ . Let's consider the following:

P(x,y) = -y

Q(x,y) = x

These functions are continuously differentiable in the region enclosed by the curve $\gamma(t)$ . Substituting these functions into Green's Theorem, we obtain:

\int_{\gamma} (-ydx + xdy) = \iint_{D} \left(\frac{\partial x}{\partial x} - \frac{\partial (-y)}{\partial y}\right) dA

Simplifying the right-hand side, we get:

\int_{\gamma} (-ydx + xdy) = \iint_{D} (1 + 1) dA

\int_{\gamma} (-ydx + xdy) = 2\iint_{D} dA

Area Relation

The left-hand side of the equation represents the line integral of the vector field $(-y, x)$ around the curve $\gamma(t)$ . This line integral can be evaluated using the parametric equations of the curve:

\int_{\gamma} (-ydx + xdy) = \int_{t=0}^{T} (-y(t) \frac{dx}{dt} + x(t) \frac{dy}{dt}) dt

Using the parametric equations of the curve, we can express $x(t)$ and $y(t)$ in terms of the parameter $t$ . After some algebraic manipulations, we can show that the line integral reduces to:

\int_{\gamma} (-ydx + xdy) = \frac{1}{2} \int_{t=0}^{T} (x(t) \frac{dy}{dt} - y(t) \frac{dx}{dt}) dt

Final Derivation

Equating the two expressions for the line integral, we obtain:

\frac{1}{2} \int_{t=0}^{T} (x(t) \frac{dy}{dt} - y(t) \frac{dx}{dt}) dt = 2\iint_{D} dA

Simplifying the right-hand side, we get:

\frac{1}{2} \int_{t=0}^{T} (x(t) \frac{dy}{dt} - y(t) \frac{dx}{dt}) dt = 2A

where $A$ represents the area enclosed by the curve $\gamma(t)$ .

Conclusion

In this article, we have demonstrated the application of Green's Theorem to derive a fundamental relation for the area enclosed by a closed curve in the plane. By choosing specific functions $P$ and $Q$ , we were able to establish a connection between the line integral of a vector field around the curve and the double integral of a function over the region enclosed by the curve. This relation provides a powerful tool for calculating the area enclosed by a closed curve in the plane.

References

Green, G. (1833). "On the laws of reflexion and refraction of light at the common surface of two non-crystallized media." Transactions of the Cambridge Philosophical Society, 7, 1-24.
Spivak, M. (1965). Calculus on Manifolds. New York: W.A. Benjamin.

Introduction

In our previous article, we explored the application of Green's Theorem to derive a fundamental relation for the area enclosed by a closed curve in the plane. This theorem has far-reaching implications in various branches of mathematics and physics. In this article, we will address some common questions and provide additional insights into the relation area inside a closed curve by Green's Theorem.

Q&A

Q: What is the significance of Green's Theorem in multivariable calculus?

A: Green's Theorem is a fundamental result in multivariable calculus that establishes a connection between line integrals and double integrals. It has numerous applications in physics, engineering, and other fields, including the calculation of area, work, and energy.

Q: What are the conditions for applying Green's Theorem?

A: Green's Theorem requires that the curve $\gamma$ be a simple closed curve, enclosing a region $D$ in the plane. The functions $P$ and $Q$ must be continuously differentiable in the region $D$ .

Q: How do I choose the functions $P$ and $Q$ for Green's Theorem?

A: The choice of functions $P$ and $Q$ depends on the specific problem you are trying to solve. In the case of calculating the area enclosed by a curve, we can choose $P(x,y) = -y$ and $Q(x,y) = x$ .

Q: What is the physical interpretation of the line integral in Green's Theorem?

A: The line integral in Green's Theorem represents the work done by a force field around a closed curve. In the case of calculating the area enclosed by a curve, the line integral represents the work done by the force field in moving a particle around the curve.

Q: Can I use Green's Theorem to calculate the area enclosed by a curve with a non-rectifiable boundary?

A: No, Green's Theorem requires that the curve $\gamma$ be a simple closed curve, enclosing a region $D$ in the plane. If the boundary of the curve is non-rectifiable, then Green's Theorem cannot be applied.

Q: How do I apply Green's Theorem to a curve with a parametric equation?

A: To apply Green's Theorem to a curve with a parametric equation, you need to express the functions $P$ and $Q$ in terms of the parameter $t$ . Then, you can use the parametric equations to evaluate the line integral.

Q: What are some common applications of Green's Theorem in physics and engineering?

A: Green's Theorem has numerous applications in physics and engineering, including:

Calculating the area enclosed by a curve
Evaluating the work done by a force field around a closed curve
Determining the energy stored in a magnetic field
Calculating the electric flux through a closed surface

Conclusion

In this article, we have addressed some common questions and provided additional insights into the relation area inside a closed curve by Green's Theorem. We hope that this Q&A article has been helpful in clarifying the application of Green's Theorem in multivariable calculus.

References

Green, G. (1833). "On the laws of reflexion and refraction of light at the common surface of two non-crystallized media." Transactions of the Cambridge Philosophical Society, 7, 1-24.
Spivak, M. (1965). Calculus on Manifolds. New York: W.A. Benjamin.