Question

Explain why the two forms of Green's Theorem are analogs of the Fundamental Theorem of Calculus.

Short answer

Explain why the two forms of Green's Theorem are analogs of the Fundamental Theorem of Calculus. The two forms of Green's Theorem are analogs of the Fundamental Theorem of Calculus because they both establish a connection between differentiation and integration in their respective domains. Green's Theorem generalizes the Fundamental Theorem of Calculus to two-dimensional vector fields. Both theorems show that integrating a derivative in some form yields a net change in the underlying function. In specific cases where curl or divergence vanishes in Green's Theorem, there is an analogous relationship to the Fundamental Theorem of Calculus concerning potential functions and vector potentials. This correspondence emphasizes the connection between differentiation and integration in both Green's Theorem and the Fundamental Theorem of Calculus.

Step-by-step solution

Understand the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus states that if a function f(x) has an antiderivative F(x) on an interval [a, b], then \[\int_a^b f(x) dx = F(b) - F(a)\] This theorem connects differentiation and integration, showing that integrating the derivative of a function over an interval yields the net change of the function over that interval.

Learn Green's Theorem

Green's Theorem is a result in vector calculus that relates the line integral of a vector field around a simple closed curve C to the double integral of the curl of the vector field over the region D enclosed by the curve C. Green's Theorem has two forms: 1. Circulation form: For a vector field \(\vec{F}(x,y) = P(x,y)\vec{i} + Q(x,y)\vec{j}\), Green's theorem states \[\oint_C (P dx + Q dy) = \iint_D (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}) dA\] 2. Flux form: For a vector field \(\vec{F}(x,y) = P(x,y)\vec{i} + Q(x,y)\vec{j}\), Green's theorem states \[\oint_C (Pdy - Qdx) = \iint_D (\frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y}) dA\]

Analog between the Fundamental Theorem of Calculus and Green's Theorem in circulation form

In circulation form, vanishing curl (\(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 0\)) implies that there is a potential function \(\phi(x,y)\) such that \[P = \frac{\partial \phi}{\partial x}\] \[Q = \frac{\partial \phi}{\partial y}\] Now, consider a line integral along curve C in the xy plane, from point A to point B, say, \[\int_A^B (P dx + Q dy)\] Since the curl vanishes, we can write the line integral as \[\int_A^B (\frac{\partial \phi}{\partial x} dx + \frac{\partial \phi}{\partial y} dy)\] This is analogous to the Fundamental Theorem of Calculus: \[\int_a^b \frac{d\phi}{dx} dx = \phi(B) - \phi(A)\]

Analog between the Fundamental Theorem of Calculus and Green's Theorem in flux form

The analog in the flux form is similar. For vanishing divergence (\(\frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} = 0\)), we can find a vector potential \(\vec{A}(x,y) = A(x,y) \vec{i} + B(x,y) \vec{j}\) such that \[P = \frac{\partial A}{\partial y} - \frac{\partial B}{\partial x} \] \[Q = \frac{\partial A}{\partial x} + \frac{\partial B}{\partial y} \] Now, consider a line integral along curve C in the xy plane, from point A to point B, say, \[\int_A^B (P dy - Q dx)\] We can write the line integral as \[\int_A^B (\frac{\partial A}{\partial y} dy - \frac{\partial A}{\partial x} dx)\] This again corresponds to the Fundamental theorem of calculus: \[\int_a^b \frac{dA}{dx} dx = A(B) - A(A)\] In conclusion, the two forms of Green's Theorem are analogs of the Fundamental Theorem of Calculus because they both show the connection between differentiation and integration in their respective domains. In the case of Green's Theorem, the relationships hold for two-dimensional vector fields and the line integrals around closed curves, whereas the Fundamental Theorem of Calculus relates the integral of a function's derivative to the function's antiderivative.