91Ó°ÊÓ

In vector calculus, a **gradient field** is a vector field that can be expressed as the gradient of some scalar function. Let's break this down. Suppose you have a scalar function, such as \( f(x, y, z) = (x^2 + y^2 + z^2)^{-1/2} \). When you compute its gradient, \( abla f \), you're essentially finding a vector field that represents the direction and rate of fastest increase of the function \( f \).
For example, the gradient of our function is:

• \( abla f = \left( -\frac{x}{(x^2 + y^2 + z^2)^{3/2}}, -\frac{y}{(x^2 + y^2 + z^2)^{3/2}}, -\frac{z}{(x^2 + y^2 + z^2)^{3/2}} \right) \)

This vector field, \( \mathbf{F} = abla f \), points in the direction of the steepest descent of the scalar field \( f \).
Intuitively, if you imagine this field as a landscape, arrows in the vector field show the steepest path downhill.

A **line integral** can be thought of as a way of summing up values along a curve. When dealing with vector fields, the line integral specifically measures the cumulative effect of a vector field along a path.
In our exercise, the curve is the circle defined by \( x^2 + y^2 = a^2 \) in the xy-plane. The path is parameterized as \( \mathbf{r}(t) = a \cos t \mathbf{i} + a \sin t \mathbf{j} \), effectively tracing out the circle as \( t \) varies from 0 to \( 2\pi \).
The **line integral** of a vector field \( \mathbf{F} = abla f \) over this path is given by:

• \( \int_0^{2\pi} \mathbf{F} \cdot d\mathbf{r} \)

By computing this, we discovered that there is no net circulation around the circle, meaning the **line integral** evaluates to zero.
This result tells us that the work done in moving a particle around this path in the field \( \mathbf{F} \) is zero.

**Stokes' Theorem** connects line integrals over a closed curve to surface integrals over a surface bounded by that curve. It essentially states that the circulation of a vector field around a closed path is equal to the sum of the curls over the surface it bounds.
Given the vector field \( \mathbf{F} = abla f \), Stokes' Theorem is used to check the circulation around our circular path by switching to a surface integral:

• \( \oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (abla \times \mathbf{F}) \cdot d\mathbf{S} \)

In simple terms, we equate a line integral around a loop to the integral of the curl over the surface.
Since \( abla \times \mathbf{F} = abla \times abla f = 0 \) (because the curl of a gradient is always zero), the surface integral also evaluates to zero.
This beautifully reinforces our previous finding that the line integral is zero.

The **curl** of a vector field measures the tendency of particles in the field to rotate around a point. It is a vector that indicates the direction and magnitude of that rotation.
For a vector field \( \mathbf{F} \), the curl is denoted by \( abla \times \mathbf{F} \). When \( \mathbf{F} = abla f \), the field is derived from a scalar potential, and its curl is always zero:

• \( abla \times abla f = 0 \)

This reflects the fact that the gradient fields are conservative, meaning there is no 'twisting' or circulation within the field.
In our exercise, the zero curl illustrates why the circulation around the circle is zero. There is no rotational component in \( \mathbf{F} \) to contribute to the path integral.
Understanding curl helps in visualizing the broader dynamics of vector fields and their potential behaviors over a given domain.