Question

Average circulation Let \(S\) be a small circular disk of radius \(R\) centered at the point \(P\) with a unit normal vector \(\mathbf{n}\). Let \(C\) be the boundary of \(S\). a. Express the average circulation of the vector field \(\mathbf{F}\) on \(S\) as a surface integral of \(\nabla \times \mathbf{F}\) b. Argue that for small \(R\), the average circulation approaches \(\left.(\nabla \times \mathbf{F})\right|_{P} \cdot \mathbf{n}(\text { the component of } \nabla \times \mathbf{F} \text { in the direction of } \mathbf{n}\) evaluated at \(P\) ) with the approximation improving as \(R \rightarrow 0\)

Short answer

Answer: The expression for the average circulation of a vector field on a small circular disk S of radius R is given by: $$\frac{1}{\pi R^2}\iint_{S} (\nabla \times \mathbf{F}) \cdot d \mathbf{A}$$ As R approaches 0, the average circulation approaches \(\left.(\nabla \times \mathbf{F})\right|_{P} \cdot \mathbf{n}\), with the approximation improving as \(R \rightarrow 0\).

Step-by-step solution

Relate circulation to the curl of a vector field

Circulation is the line integral of a vector field \(\mathbf{F}\) around a closed curve \(C\). For this particular setup, the circulation is: $$\oint_{C} \mathbf{F} \cdot d\mathbf{r}$$ By Stokes' theorem, this circulation is equal to the surface integral of the curl of the vector field over the surface \(S\): $$\oint_{C} \mathbf{F} \cdot d\mathbf{r} = \iint_{S} (\nabla \times \mathbf{F}) \cdot d \mathbf{A}$$

Express the average circulation as a surface integral

To find the average circulation over surface \(S\), we'll first find the area of \(S\): $$A = \pi R^2$$ Now, divide the total circulation (the surface integral) by the area of \(S\) to obtain the average circulation: $$\text{Average Circulation} = \frac{1}{A}\iint_{S} (\nabla \times \mathbf{F}) \cdot d \mathbf{A}$$

Find the limit of the average circulation as R approaches 0

As \(R\) approaches 0, the enclosed area \(A\) approaches 0 as well. To find the limit, we'll express the average circulation using the definition of the limit: $$\lim_{R \rightarrow 0} \frac{1}{\pi R^2}\iint_{S} (\nabla \times \mathbf{F}) \cdot d \mathbf{A}$$ When \(R\) is very small, the curl of \(\mathbf{F}\), or \(\nabla \times \mathbf{F}\), will be nearly constant over the small area \(S\). Therefore, as \(R \rightarrow 0\), the average circulation will approach \(\left.(\nabla \times \mathbf{F})\right|_{P} \cdot \mathbf{n}\). In summary: a. The average circulation of the vector field \(\mathbf{F}\) on \(S\) can be expressed as the surface integral: $$\frac{1}{\pi R^2}\iint_{S} (\nabla \times \mathbf{F}) \cdot d \mathbf{A}$$ b. For small \(R\), the average circulation approaches \(\left.(\nabla \times \mathbf{F})\right|_{P} \cdot \mathbf{n}\), with the approximation improving as \(R \rightarrow 0\).

Key concepts

Stokes' theorem

Stokes' theorem is a powerful tool in vector calculus that helps connect surface integrals to line integrals. It allows us to relate the circulation of a vector field over a closed curve to the curl of the field over the surface bounded by that curve.
Stokes' theorem can be formally stated as:

• \(\oint_{C} \mathbf{F} \cdot d\mathbf{r} = \iint_{S} (abla \times \mathbf{F}) \cdot d \mathbf{A} \)

Here, \(C\) is the boundary of the surface \(S\), and \(d\mathbf{A}\) is a vector normal to the surface \(S\).
This theorem is extensively used to transform complicated line integrals into often simpler surface integrals. In practical terms, this shortcut is particularly useful when the geometry of the setup is such that calculating the curl over the surface is straightforward.
Let's apply it to our exercise: the circulation around the boundary \(C\) becomes the surface integral over the curl of \(\mathbf{F}\) over \(S\). It connects the motion and rotation (described by circulation) to the vector field's rotational tendency (described by its curl).

curl of a vector field

The curl of a vector field is an essential concept in understanding the rotational effects of a vector field in a fluid or an electromagnetic context. To put it simply, the curl measures the tendency of the field to rotate around a point or line.
Mathematically, it is expressed as:

• \(abla \times \mathbf{F} = \mathbf{curl} \, \mathbf{F}\)

The components of \(abla \times \mathbf{F}\) can be found using the determinant of a special matrix, incorporating partial derivatives and unit vectors for each axis direction.
In our exercise, the focus is on calculating this curl and how it affects circulation around a boundary. As \(R \rightarrow 0\), the curl is essentially uniform over the disk \(S\), thus helping us to determine how the circulation becomes proportional to the curl of \(\mathbf{F}\) in the direction of the normal vector \(\mathbf{n}\) at point \(P\).
The curl, thus, gives us a local representation of how much \(\mathbf{F}\) swirls around \(P\), providing insights into fluid dynamics or electromagnetic fields in those small zones.

average circulation

The concept of average circulation in a vector field means evaluating how the field circulates per unit area on a given surface. For any surface \(S\), the average circulation gives us an idea about the overall effect rather than just a localized situation.
In our exercise, we calculate the average using the formula:

• \(\text{Average Circulation} = \frac{1}{\pi R^2}\iint_{S} (abla \times \mathbf{F}) \cdot d \mathbf{A}\)

As \(R\) approaches 0, the average circulation converges to the vector dot product \((abla \times \mathbf{F}) \cdot \mathbf{n}\).
This result shows how the circulation depends on the component of the vector field's curl along the direction of the normal vector \(\mathbf{n}\). This reasoning forms the basis for understanding field dynamics as observed from the small-scale behavior of the vector field \(\mathbf{F}\).
In summary, as the radius \(R\) is squeezed to an infinitesimally small value, the average circulation becomes a direct reflection of the vector field's rotational characteristic at that point, making \(\text{\((abla \times \mathbf{F}) \cdot \mathbf{n}\)}\) a vital observation tool in applied physics and engineering.