Question

Verify Green's Theorem by using a computer algebra system to evaluate both integrals \(\int_{C} x e^{y} d x+e^{x} d y=\int_{R} \int\left(\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}\right) d A\) for the given path. \(C:\) boundary of the region lying between the graphs of \(y=x\) and \(y=x^{3}\) in the first quadrant

Short answer

This result of this exercise is that the two evaluated integrals are indeed equal, thus verifying the Green's theorem for the provided vector field along path \(C\). The actual computation can involve complex calculations, so a computer algebra system would be beneficial.

Step-by-step solution

Identify the Functions

First, identify the functions in the integrals. The line integral is given by \( Mdx + Ndy \) with M(x,y) = \(x e^{y}\) and N(x,y) = \(e^{x}\). Therefore, \( \frac{\partial M}{\partial y} = x e^{y} \) and \( \frac{\partial N}{\partial x} = e^{x} \)

Perform the Line Integral

Next, compute the line integral over curve C. To do this, parameterize the curve C. In this case, we have two equations which define the limits of the region: \(y = x\) and \(y = x^{3}\). As for \(x\), it goes from 0 to 1. The actual computation of the integral can be complicated, so it's recommended to use a computer algebra system.

Compute the Double Integral

Now compute the double integral \( \int_R \int (\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}) dA \). Here, \(R\) is the region enclosed by \(C\), with bounds \(0 \leq x \leq 1\) and \(x^{3} \leq y \leq x\). Again, it's recommended to use a computer algebra system to compute this.

Compare the Results

Finally, compare the two results. Greens' Theorem will hold true if they are equal, thereby verifying the theorem.

Key concepts

Line Integral

Line integrals are a fundamental concept in vector calculus, particularly when analyzing the flow across a curve or the work done by a force field along a path. In essence, a line integral sums up a function along a curve, giving you a single scalar value.
Consider the general form of line integrals for functions in two dimensions, expressed as \( \( \int_{C} M dx + N dy \) \), where \( C \) is a smooth curve, \( M(x,y) \) and \( N(x,y) \) are the component functions of a vector field, and \( dx \) and \( dy \) signify infinitesimal displacements along the \( x \) and \( y \) axes respectively.
When evaluating a line integral, you're often asked to parameterize the curve \( C \) based on the coordinates \( x \) and \( y \) as functions of a parameter \( t \) that varies over a certain interval. For example, in our exercise with the region between the graphs \( y = x \) and \( y = x^3 \) in the first quadrant, we would express \( x \) and \( y \) in terms of a parameter that reflects these boundaries, typically choosing \( t \) to vary from 0 to 1. From there, we would substitute these parameterized expressions into the integral and solve, a process that can be greatly facilitated by a computer algebra system.

Importance in Green's Theorem

Green's Theorem connects line integrals around a simple closed curve \( C \) with a double integral over the plane region \( R \) that \( C \) encloses. The theorem essentially states that the line integral around the boundary of \( R \) is equal to the double integral over \( R \) of the curl of \( \Vec{F} \) (a vector field composed of \( M \) and \( N \) in our case).

Double Integral

Double integrals extend the concept of a single integral to functions of two variables over a two-dimensional region. They tally up the volume under a surface over a specified area, and in the context of Green's Theorem, they represent the flux of a vector field through the region.
Given an area \( R \) in the plane, the double integral \( \int_R \int f(x,y) dA \) is evaluated over \( R \), where \( dA \) is an infinitesimal area element and \( f(x,y) \) is a scalar function. To compute this, you often set up the double integral as an iterated integral, integrating first with respect to \( y \) and then with respect to \( x \) (or vice versa), respecting the bounds of the region \( R \) throughout.

Application in Green's Theorem

In the exercise, the double integral computes the curl of the vector field created by the functions \( M(x,y) = x e^y \) and \( N(x,y) = e^x \) over region \( R \) (the area between the curves \( y = x \) and \( y = x^3 \) in the first quadrant). According to Green's Theorem, computing this area integral of the curl gives us the same result as the circulation of the vector field along the boundary \( C \) – that is, the line integral covered in the previous section.

Partial Derivatives

Partial derivatives play a vital role in multivariable calculus, particularly when dealing with functions of several variables. They represent the rate at which a function changes as one variable shifts while all others remain fixed. Symbolically, the partial derivative of a function \( f(x,y) \) with respect to \( x \) is denoted by \( \frac{\partial f}{\partial x} \) and provides the slope of the tangent line to the curve on the \( f(x,y) \) surface resulting from holding \( y \) constant. Similarly, \( \frac{\partial f}{\partial y} \) gives the slope with respect to changes in \( y \) while holding \( x \) constant.
In our exercise, we focus on \( M(x,y) = x e^y \) and \( N(x,y) = e^x \) and their partial derivatives with respect to \( y \) and \( x \) respectively. Calculating these, we get \( \frac{\partial M}{\partial y} = x e^y \) and \( \frac{\partial N}{\partial x} = e^x \) as indicated in the first step of the step-by-step solution.

Relevance to Green's Theorem

The double integral in Green's Theorem involves the difference of these partial derivatives, \( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \). This difference is key to establishing the connection between the circulation of a vector field along a closed curve and the flux of the field across the area it encloses. The equation \( \int_{C} Mdx + Ndy = \int_R \int (\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}) dA \) can thus be verified by showing that both integrals yield the same result, providing a concrete application of partial derivatives to verify Green's Theorem.