91Ó°ÊÓ

In vector calculus, a potential function is a scalar function whose gradient yields a given vector field. If you have a vector field \( \mathbf{F} = abla f \), then \( f \) is considered the potential function of \( \mathbf{F} \). To find the potential function, you need to identify the function \( f \) such that the gradient \( abla f \) matches the components of \( \mathbf{F} \). This essentially involves integration of the vector field components.

In the exercise, the vector field \( \mathbf{F}(x, y, z) = yz \mathbf{i} + xz \mathbf{j} + (xy + 2z) \mathbf{k} \) needed to be matched with a gradient. Each component of \( \mathbf{F} \) was integrated with respect to its variable. This produced part of the potential function:

• Integrating \( yz \) with respect to \( x \) resulted in \( xyz + g(y, z) \).
• Integrating \( xz \) with respect to \( y \) ensured compatibility with other components.
• Integrating \( (xy + 2z) \) with respect to \( z \) completed the function as \( xyz + z^2 + C \).

The final result, \( f(x, y, z) = xyz + z^2 + C \), is the desired potential function consistent with \( \mathbf{F} = abla f \). The constant \( C \) arises as integration constants, as gradient fields only determine the potential function up to an additive constant.

A line integral, in the context of vector fields, measures the magnitude of a vector field along a curve. This is akin to accumulating tiny bits of the vector field's influence as one moves along the curve

• The integral \( \int_{C} \mathbf{F} \cdot d \mathbf{r} \) captures the total effect of \( \mathbf{F} \) as it passes along the path \( C \).
• A line integral takes into account not only the path \( C \) itself but also the inclination of \( \mathbf{F} \) relative to \( C \).

To find a line integral, you traditionally parameterize the path \( C \) and evaluate \( \mathbf{F} \cdot d\mathbf{r} \) along it. However, when \( \mathbf{F} \) is conservative, this calculation can be greatly simplified.

In our exercise, instead of following the traditional path-dependent method, we were able to utilize the potential function to determine the integral across \( C \) by only considering the endpoints, thanks to the next concept: the fundamental theorem for line integrals.

The fundamental theorem for line integrals elegantly simplifies the computation of line integrals in conservative vector fields. When a vector field \( \mathbf{F} \) is conservative, this theorem states that the line integral along a path \( C \) can be calculated simply as the difference of the potential function \( f \) between the endpoints of the path:

• \( \int_{C} \mathbf{F} \cdot d\mathbf{r} = f(B) - f(A) \), where \( A \) and \( B \) are start and end points on \( C \).

This theorem is extremely powerful as it eliminates the need to perform tedious parameterization and integration along the path.

In the exercise, the path was a straight line segment from \( (1,0,-2) \) to \( (4,6,3) \). By applying the fundamental theorem, we computed the potential function at these points:

• \( f(4,6,3) = 81 \)
• \( f(1,0,-2) = 4 \)

Taking their difference, \( 81 - 4 = 77 \), provided the value of the line integral \( \int_{C} \mathbf{F} \cdot d\mathbf{r} \). This direct method showcases the convenience offered by the theorem in dealing with conservative fields.