Problem 131 - Calculus Volume 3

Question

For the following exercises, $$ \mathbf{F}(x, y)=2 x y^{2} \mathbf{i}+\left(2 y x^{2}+2 y\right) \mathbf{j} $$ \(G(x, y)=(y+x) \mathbf{i}+(y-x) \mathbf{j}, \quad\) and let \(C_{1}\) be the curve consisting of the circle of radius 2 , centered at the origin and oriented counterclockwise, and \(C_{2}\) be the curve consisting of a line segment from (0,0) to (1,1) followed by a line segment from (1,1) to (3,1) . Calculate the line integral of \(\mathbf{F}\) over \(C_{1}\)

Step-by-Step Solution

Answer

The line integral of \( \mathbf{F} \) over \( C_1 \) is 0.

Step 1: Convert the integral into a known form

We need to calculate the line integral \( \oint_{C_1} \mathbf{F} \cdot d\mathbf{r} \), where \( C_1 \) is the circle with radius 2 centered at the origin. For such conservative fields, the line integral over any closed path is zero according to Green's Theorem if \( abla \times \mathbf{F} = 0 \). We check if \( \mathbf{F} \) is conservative by ensuring \( abla \times \mathbf{F} = 0 \).

Step 2: Calculate the curl of the vector field \( \mathbf{F} \)

The vector field \( \mathbf{F} = 2xy^2 \mathbf{i} + (2yx^2 + 2y) \mathbf{j} \). The curl in two dimensions \((x, y)\) is computed as: \[ abla \times \mathbf{F} = \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \] where \( P = 2xy^2 \) and \( Q = 2yx^2 + 2y \). Compute: \( \frac{\partial Q}{\partial x} = 4xy \) and \( \frac{\partial P}{\partial y} = 4xy \). Thus, \( abla \times \mathbf{F} = 0 \).

Step 3: Applying Green's Theorem

Since \( abla \times \mathbf{F} = 0 \), \( \mathbf{F} \) is conservative. According to Green's Theorem, the line integral of a conservative vector field over a closed path \( C_1 \) is zero, i.e., \( \oint_{C_1} \mathbf{F} \cdot d\mathbf{r} = 0 \).

Key Concepts

Green's Theorem

Green's Theorem is a powerful tool in vector calculus. It allows us to connect a line integral around a simple closed curve to a double integral over the region it encloses. If you have a vector field \( \mathbf{F} = P(x, y) \mathbf{i} + Q(x, y) \mathbf{j} \), Green's Theorem states that the line integral around a positively oriented, piecewise smooth, simple closed curve \( C \) is equal to the double integral over the region \( D \) it encloses:

• \[ \oint_{C} \mathbf{F} \cdot d\mathbf{r} = \iint_{D} \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA \]

For this theorem to hold, the functions \( P \) and \( Q \) must have continuous partial derivatives on an open region that contains \( D \). This transformation from a line integral to a double integral is crucial for simplifying the calculation of line integrals over closed curves. It's especially convenient when dealing with conservative vector fields, where this integral is zero, as illustrated in the given exercise.

Conservative Vector Fields

Conservative vector fields are special because they have the property that the line integral over any closed path is zero. This means there's no "net work" done when moving around a closed loop in the field. A vector field \( \mathbf{F} \) is conservative if it can be expressed as the gradient of some scalar function \( f \), such that \( \mathbf{F} = abla f \). To check if a vector field is conservative, especially in two dimensions, you often compute the curl and verify if it is zero:

• The curl \( abla \times \mathbf{F} = 0 \) indicates a vector field is conservative.

In our exercise, upon computing the curl, it was indeed zero, confirming that \( \mathbf{F} \) is conservative. Thus, applying Green's Theorem directly led to the result that the line integral over the closed curve \( C_1 \) is zero.

Curl of a Vector Field

The curl of a vector field is a crucial concept in vector calculus, representing the rotation or "twisting" of a field in three dimensions. In two dimensions, the curl simplifies to a scalar. Given a vector field \( \mathbf{F} = P(x, y) \mathbf{i} + Q(x, y) \mathbf{j} \), the curl is defined as:

• \[ abla \times \mathbf{F} = \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \]

In the exercise, this calculation verifies if \( \mathbf{F} \) is conservative. If \( abla \times \mathbf{F} = 0 \), the vector field is conservative and path-independent, which means the total "twisting" or rotation effect around a closed loop is zero. Recognizing a zero curl was essential to applying Green's Theorem effectively in the exercise, allowing us to conclude the integral result without directly computing the path integral.