Problem 25 - Calculus Early Transcendentals

Question

Suppose that \(\mathbf{F}(x, y)=f(x, y) \mathbf{i}+g(x, y) \mathbf{j}\) is a vector field whose component functions \(f\) and \(g\) have continuous first partial derivatives. Let \(C\) denote a simple, closed, piecewise smooth curve oriented counterclockwise that bounds a region \(R\) contained in the domain of \(\mathbf{F}\). We can think of \(\mathbf{F}\) as a vector field in 3 -space by writing it as $$ \mathbf{F}(x, y, z)=f(x, y) \mathbf{i}+g(x, y) \mathbf{j}+0 \mathbf{k} $$ With this convention, explain why $$ \int_{C} \mathbf{F} \cdot d \mathbf{r}=\iint_{R} \operatorname{curl} \mathbf{F} \cdot \mathbf{k} d A $$

Step-by-Step Solution

Answer

The equality follows directly from Green's Theorem relating line integrals and curls over regions.

Step 1: Express the Line Integral

The line integral of the vector field \( \mathbf{F} \) over the curve \( C \) is given by the expression \( \int_{C} \mathbf{F} \cdot d \mathbf{r} \). Here, \( d \mathbf{r} \) is the differential vector along the curve \( C \), which is \( d \mathbf{r} = dx \mathbf{i} + dy \mathbf{j} \). Consequently, the line integral becomes \( \int_{C} (f(x, y) \, dx + g(x, y) \, dy) \).

Step 2: Define the Region and the Concept of Curl

The curve \( C \) is a simple, closed, piecewise smooth curve enclosing a region \( R \). According to the definition of the curl in 3-dimensional space, the curl of \( \mathbf{F} \) in this scenario is defined as \( \text{curl} \, \mathbf{F} = \left( 0 \right) \mathbf{i} + \left( 0 \right) \mathbf{j} + \left( \frac{\partial g}{\partial x} - \frac{\partial f}{\partial y} \right) \mathbf{k} \).

Step 3: Application of Green's Theorem

Green's Theorem states that for a continuously differentiable vector field \( \mathbf{F} = f(x, y) \mathbf{i} + g(x, y) \mathbf{j} \) on a region \( R \), bounded by a positively oriented, simple, closed curve \( C \), the line integral around \( C \) is equal to the double integral of the curl of \( \mathbf{F} \) over the region \( R \). Mathematically, this is expressed as \( \int_{C} \mathbf{F} \cdot d\mathbf{r} = \iint_{R} \left( \frac{\partial g}{\partial x} - \frac{\partial f}{\partial y} \right) \, dA \).

Step 4: Relate Curl to Green's Theorem Result

When applying Green's Theorem to \( \mathbf{F}(x, y) = f(x, y) \mathbf{i} + g(x, y) \mathbf{j} + 0 \mathbf{k} \), the components are such that \( \text{curl} \, \mathbf{F} \cdot \mathbf{k} = \frac{\partial g}{\partial x} - \frac{\partial f}{\partial y} \), leading to \( \iint_{R} \text{curl} \, \mathbf{F} \cdot \mathbf{k} \, \dA = \iint_{R} \left( \frac{\partial g}{\partial x} - \frac{\partial f}{\partial y} \right) \, dA \).

Step 5: Conclusion

Combining the results from Green's Theorem and the definition of curl, we have determined that \( \int_{C} \mathbf{F} \cdot d \mathbf{r} = \iint_{R} \text{curl} \, \mathbf{F} \cdot \mathbf{k} \, dA \). This equality shows the connection between a line integral over \( C \) and a surface integral over the region \( R \) using the concept of curl and Green's Theorem.

Key Concepts

Line Integral

A line integral is a way to integrate functions along a curve. It's like summing up tiny little pieces of a function over a path. When dealing with a vector field, we often compute the line integral of the vector field along a curve. This gives us the "effect" of the vector field along the path. To compute this, we use the expression \( \int_{C} \mathbf{F} \cdot d \mathbf{r} \).

Here, \( \mathbf{F} \) is a vector field and \( d \mathbf{r} \) represents a tiny segment along the curve \( C \), which is expressed as \( d \mathbf{r} = dx \mathbf{i} + dy \mathbf{j} \).

For a two-dimensional vector field, the line integral converts to \( \int_{C} (f(x, y) \, dx + g(x, y) \, dy) \). This allows us to analyze how much of the vector field "flows" along the path.

Curl

The concept of curl in vector calculus measures the rotation or swirling strength of a vector field at a point. In three-dimensional space, the curl relates to a vector field's tendency to induce rotation around a point.

For a 2D vector field \( \mathbf{F} = f(x, y) \mathbf{i} + g(x, y) \mathbf{j} \), the curl is treated as a 3D vector by introducing a "zero" component in the \( \mathbf{k} \) direction, making it \( \mathbf{F}(x, y, z) = f(x, y) \mathbf{i} + g(x, y) \mathbf{j} + 0 \mathbf{k} \).

The curl of \( \mathbf{F} \) is then the vector \( (0) \mathbf{i} + (0) \mathbf{j} + \left( \frac{\partial g}{\partial x} - \frac{\partial f}{\partial y} \right) \mathbf{k} \). The critical part is \( \left( \frac{\partial g}{\partial x} - \frac{\partial f}{\partial y} \right) \), which indicates how the vector field tends to "curl" around points in 2D.

Vector Field

A vector field assigns a vector to each point in a subset of space. Imagine putting an arrow on every point in a plane. These arrows describe how, at that point, something is "moving" or changing—like the wind's direction and speed over a region.

In our case, a vector field \( \mathbf{F} \) has the form \( \mathbf{F}(x, y) = f(x, y) \mathbf{i} + g(x, y) \mathbf{j} \). This means each point \( (x, y) \) in space has a vector determined by the functions \( f(x, y) \) and \( g(x, y) \).

A well-defined vector field must have continuous functions with first partial derivatives, ensuring smooth transitions across points. In applying Green's Theorem, this smoothness allows us to relate line integrals over a boundary to a double integral over the area inside, giving insights into curving, rotation, and overall flow across regions.