Question

Referring to both forms of Green's Theorem, match each idea in Column 1 to an idea in Column 2: $$\begin{array}{|l|l|}\hline \text { Line integral for flux } & \text { Double integral of the curl } \\\\\hline \text { Line integral for circulation } & \text { Double integral of the divergence } \\\\\hline\end{array}$$

Short answer

Answer: The line integral for circulation matches with the double integral of the curl.

Step-by-step solution

Line integral for flux — Double integral of the divergence

The line integral for flux computes the net amount of a vector field "flowing" through a curve. This is related to the double integral of the divergence of a vector field from Green's second theorem. In other words, the line integral for flux matches with the double integral of the divergence.

Line integral for circulation — Double integral of the curl

The line integral for circulation computes the amount of "swirling" or "rotation" in a vector field around a closed curve. This is related to the double integral of the curl of a vector field from Green's first theorem. In other words, the line integral for circulation matches with the double integral of the curl. So, the matching ideas are: - Line integral for flux: Double integral of the divergence - Line integral for circulation: Double integral of the curl

Key concepts

Line Integral for Flux

Understanding the line integral for flux goes a long way in comprehending how a vector field interacts with a curve. Imagine a fluid flowing through space, and let's consider a curve that acts like a fence. The line integral for flux measures how much fluid crosses that fence. It's like putting a net across a river and measuring how much water gets caught.

The mathematical expression for this idea is given by \( \oint_C \mathbf{F} \cdot d\mathbf{n} \) where \( \mathbf{F} \) is the vector field and \( d\mathbf{n} \) is a vector that points outward and is perpendicular to the curve \( C \). Here, \( \cdot \) stands for the dot product, which quantifies the component of \( \mathbf{F} \) across the curve.

To improve understanding, picture the flux integral as a toll booth recording the number of cars passing per hour. Each car's direction and speed can represent a vector, and the toll booth encapsulates the 'curve'.

Double Integral of Divergence

The double integral of the divergence is fascinating as it captures how a vector field behaves over an entire region, not just along a curve. Divergence measures the 'outflow' of the field at each point—roughly, whether the field is a source (creating flow) or a sink (absorbing flow) in a certain region.

The divergence is computed using the del operator \( abla \) and taking the dot product with the field: \( \text{div }\mathbf{F} = abla \cdot \mathbf{F} \). To capture this over a whole area, we integrate: \( \iint_R (abla \cdot \mathbf{F}) dA \), where \( R \) is the region of interest.

For an analogy, think of the area as a sponge, and the water flow through it is our vector field. Sponges have parts that absorb water and parts that squeeze water out—this integral is like adding up all those tiny contributions across the sponge's volume.

Line Integral for Circulation

The line integral for circulation gives us insight into the rotational aspect of a vector field along a closed loop. Imagine a leaf being carried by a whirlpool; this integral tells us how much the whirlpool is spinning the leaf around as it goes along a path.

This integral is visually represented by \( \oint_C \mathbf{F} \cdot d\mathbf{s} \), where \( C \) is our path and \( d\mathbf{s} \) is a tiny segment of the path, representing our leaf. The vector field \( \mathbf{F} \) is the whirlpool's currents, creating rotation around \( C \).

The rotation of air around the center of a tornado is a natural example of circulation. The tornado's wind (our vector field) causes significant circulation measurable by this line integral along a path encircling the tornado.

Double Integral of the Curl

Whereas the line integral for circulation looks at rotation around a path, the double integral of the curl quantifies rotation over an area. Curl refers to the spinning motion at a point in the field, much like the vortices you see when stirring a cup of tea.

The mathematical side of this concept is expressed through another use of the del operator \( abla \) by taking the cross product with the vector field: \( \text{curl }\mathbf{F} = abla \times \mathbf{F} \). The integral \( \iint_R (abla \times \mathbf{F}) \cdot d\mathbf{A} \) sums up these tiny twirls across an entire area \( R \).

To help visualize this, imagine that our cup of tea from earlier is now a large pond. Instead of one spoon stirring, imagine many small whirlpools scattered throughout the pond. The double integral of the curl helps us understand the cumulative effect of all these miniature swirls over the pond's surface.