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Chapter 13: Problem 23

State Green's Theorem.

Short Answer

Expert verified
Green's Theorem states: \[\oint_{C} (L dx + M dy) = \iint_{D} \left(\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y}\right) dA\] This theorem links the line integral over a positively oriented, piecewise smooth simple closed curve C to a double integral over the plane region D bounded by C.

Step by step solution

01

Stating the theorem in vector form

State Green's theorem as: if L and M are functions of (x, y) defined on an open region containing D and have continuous first partial derivatives there, then \[ \oint_{C} (L dx + M dy) = \iint_{D} \left(\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y}\right) dA \]
02

Understanding the left hand side of the theorem

The left side of the equation is a line integral over a positively oriented, piecewise smooth simple close curve C. This means we integrate the function L dx + M dy around the curve C.
03

Understanding the right hand side of the theorem

The right side of the equation is a double integral over the divergence of F, dA. This means we integrate the function \(\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y}\) over the region D.
04

Understanding the theorem as a whole

Green's theorem states that these two integrals are equal for a region D bounded by the curve C, under the conditions stated in step 1. Hence, given a field F described by the functions L and M, we can choose to perform either the line integral or the double integral based on the convenience to find the same result.

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Most popular questions from this chapter

Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) where \(C\) is represented by \(\mathbf{r}(t)\) \(\mathbf{F}(x, y)=3 x \mathbf{i}+4 y \mathbf{j}\) \(\quad C: \mathbf{r}(t)=t \mathbf{i}+\sqrt{4-t^{2}} \mathbf{j}, \quad-2 \leq t \leq 2\)

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In Exercises 41 and \(42,\) evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) for each curve. Discuss the orientation of the curve and its effect on the value of the integral. \(\mathbf{F}(x, y)=x^{2} \mathbf{i}+x y \mathbf{j}\) (a) \(\mathbf{r}_{1}(t)=2 t \mathbf{i}+(t-1) \mathbf{j}, \quad 1 \leq t \leq 3\) (b) \(\mathbf{r}_{2}(t)=2(3-t) \mathbf{i}+(2-t) \mathbf{j}, \quad 0 \leq t \leq 2\)

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