Question

Green's Theorem as a Fundamental Theorem of Calculus Show that if the circulation form of Green's Theorem is applied to the vector field \(\left\langle 0, \frac{f(x)}{c}\right\rangle\), where \(c>0\) and \(R=\\{(x, y): a \leq x \leq b, 0 \leq y \leq c\\},\) then the result is the Fundamental Theorem of Calculus, $$ \int_{a}^{b} \frac{d f}{d x} d x=f(b)-f(a) $$

Short answer

In this solution, we demonstrated that the circulation form of Green's Theorem can be used to derive the Fundamental Theorem of Calculus. By considering a vector field \(\textbf{F}\) with the components \(\left\langle 0, \frac{f(x)}{c}\right\rangle\), we first computed the curl of the vector field. Then we performed the line integral along the boundary of the rectangular region, and the double integral over the region. By applying Green's theorem, we found that the line integral was equivalent to the double integral, leading us to the Fundamental Theorem of Calculus, which states \(\int_{a}^{b} \frac{d f}{d x} dx = f(b) - f(a)\).

Step-by-step solution

1. Compute the curl of the vector field.

First, we need to compute the curl of the vector field \(\textbf{F}\). The curl is given by: $$ \nabla \times \textbf{F} = \left(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}\right) \times \left\langle 0, \frac{f(x)}{c}, 0\right\rangle $$ The only non-zero component of the curl is the one in the z-direction, which is given by: $$ (\nabla \times \textbf{F})_z = \frac{\partial}{\partial x}\left(\frac{f(x)}{c}\right) - \frac{\partial}{\partial y}(0) = \frac{1}{c}\frac{d f}{d x} $$ So the curl of \(\textbf{F}\) is: $$ \nabla \times \textbf{F} = \left\langle 0, 0, \frac{1}{c}\frac{d f}{d x}\right\rangle $$

2. Perform the line integral.

Now we need to perform the line integral of the vector field \(\textbf{F}\) around the boundary of the region \(R\). We can split this integral into four parts, corresponding to the four sides of the rectangle: $$ \oint_C \textbf{F} \cdot d\textbf{r} = \int_{0}^{c} \textbf{F}(a,y) \cdot \left\langle 0, dy\right\rangle + \int_{a}^{b} \textbf{F}(x,c) \cdot \left\langle dx, 0\right\rangle + \int_{0}^{c} \textbf{F}(b,y) \cdot \left\langle 0, dy\right\rangle + \int_{a}^{b} \textbf{F}(x,0) \cdot \left\langle dx, 0\right\rangle $$ However, since \(\textbf{F}(x,0) = \left\langle 0, 0\right\rangle\) and \(\textbf{F}(x,c) = \left\langle 0, \frac{f(x)}{c}\right\rangle\), the only non-zero contribution comes from the first and third integrals. So we have: $$ \oint_C \textbf{F} \cdot d\textbf{r} = \int_{0}^{c} \frac{f(a)}{c}dy - \int_{0}^{c} \frac{f(b)}{c}dy = cf(a) - cf(b) $$

3. Perform the double integral.

Next, we need to compute the double integral of the curl over the region \(R\): $$ \iint_R (\nabla \times \textbf{F})\cdot \textbf{k} \, dA = \iint_R \frac{1}{c}\frac{d f}{d x} dA = \frac{1}{c} \int_{a}^{b} \int_{0}^{c} \frac{d f}{d x} dy dx $$ Since the inside integral is with respect to \(y\), we can treat \(\frac{d f}{d x}\) as a constant: $$ \frac{1}{c} \int_{a}^{b} \int_{0}^{c} \frac{d f}{d x} dy dx = \frac{1}{c} \int_{a}^{b} \frac{d f}{d x} \int_{0}^{c} dy dx $$ Now we can perform the inside integral: $$ \frac{1}{c}\int_{a}^{b} \frac{d f}{d x} \int_{0}^{c} dy dx = \frac{1}{c}\int_{a}^{b} \frac{d f}{d x} c dx = \int_{a}^{b} \frac{d f}{d x} dx $$

4. Use Green's theorem.

Now that we have computed both sides of Green's theorem, we can put them together: $$ \oint_C \textbf{F} \cdot d\textbf{r} = \iint_R (\nabla \times \textbf{F})\cdot \textbf{k} \, dA $$ This gives us: $$ cf(a) - cf(b) = \int_{a}^{b} \frac{d f}{d x} dx $$ Dividing both sides by \(c\) and rearranging terms, we obtain the Fundamental Theorem of Calculus: $$ \int_{a}^{b} \frac{d f}{d x} dx = f(b) - f(a) $$

Key concepts

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects the concept of differentiation with integration. It states that if a function is continuous over an interval and has an antiderivative, then the definite integral of its derivative over the interval gives the difference in the values of the antiderivative at the endpoints.

The theorem can be expressed as:

• If you have a function, say function \( f \), defined on a closed interval \( [a, b] \).
• And if \( F \) is an antiderivative of \( f \) on \( [a, b] \),
• Then, the definite integral of \( f \) from \( a \) to \( b \) is given by: \( \int_{a}^{b} f(x) \, dx = F(b) - F(a) \).

This property is profoundly used in solving problems involving area and helps in evaluating the total accumulated quantity like distance, area, and volume.

Vector Field

A vector field is essentially a function that assigns a vector to every point in space. Imagine a field of arrows spread across a plane, where each arrow has a magnitude and direction based on a corresponding vector.

For example, consider the vector field \( \mathbf{F}(x, y) = \left\langle 0, \frac{f(x)}{c} \right\rangle \). Here, the field defined in a two-dimensional space only has a nonzero component in the y-direction. This means every vector in the field points either up or down (vertically) depending on the sign of \( \frac{f(x)}{c} \).

• Vector fields can represent different physical phenomena, such as gravitational fields or velocity fields in fluid dynamics.
• In our context, these fields are central to applying Green's Theorem to transform a line integral around a closed loop into a double integral over the plane region it encloses.

Curl

The curl of a vector field is a measure of its rotation or the tendency for the field's vectors to "curl" around a point. It's a vector itself and provides information about the local rotational effect of a vector field.

In three-dimensional space, the curl is defined using the cross product of the del operator:

• For the vector field \( \mathbf{F}(x, y, z) = \left\langle P(x,y,z), Q(x,y,z), R(x,y,z) \right\rangle \), the curl \( abla \times \mathbf{F} \) is given by:

\[ abla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}, \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right)\]For the vector field \( \left\langle 0, \frac{f(x)}{c}, 0 \right\rangle \), only the \( z \)-component of the curl is non-zero, and it simplifies to \( \frac{1}{c} \frac{d f}{d x} \) which helps transition into evaluating the integral needed for Green's Theorem.

Curl is crucial to Green's Theorem as it involves finding the circulation of a vector field, connecting it to the region's enclosed boundary and its divergence.

Line Integral

A line integral is a type of integration used to calculate the total of a field along a curve. It essentially sums up the field values weighted by their direction along a path.

Green's Theorem applies line integrals to transform them into double integrals over a plane region, making evaluations simpler for certain problems.

• To perform a line integral \( \oint_C \textbf{F} \cdot d\textbf{r} \), we integrate the vector field \( \mathbf{F} \) along a closed curve \( C \).
• In the context of Green's Theorem, it's useful for evaluating circulation or the total flow along a boundary.
• This involves parameterizing the boundary and computing the weighted sum of the vector field along the perimeter.

In our solution, the line integral around rectangle \( R \) simplifies greatly, with only certain edges contributing, leading to an expression that aligns with the application of the Fundamental Theorem of Calculus.