91Ó°ÊÓ

In vector calculus, the concept of a gradient vector field is crucial for understanding how scalar potential functions relate to vector fields. When you have a scalar function denoted as \( f(x, y, z) \), the gradient \( abla f \) represents a vector field. This is because the gradient of \( f \) is composed of the partial derivatives of \( f \) with respect to each of the variables \( x, y, \) and \( z \). The result is:

• The function \( \frac{\partial f}{\partial x} \) which tells us how \( f \) changes as \( x \) changes, with \( y \) and \( z \) held constant.
• \( \frac{\partial f}{\partial y} \) which describes the change in \( f \) when \( y \) changes, while \( x \) and \( z \) are fixed.
• \( \frac{\partial f}{\partial z} \) which indicates the variation in \( f \) with \( z \).

The process involves solving these partial differential equations to determine \( f(x, y, z) \). In our problem, the gradient vector field was \( \mathbf{F} = (e^y, xe^y, (z+1)e^z) \), from which we derived the potential function \( f(x, y, z) = xe^y + ze^z + C \). Understanding these concepts help us see how the configuration of scalar fields can influence and determine vector fields.

A line integral in vector calculus is a way to integrate functions along a curve. Specifically, it calculates the accumulation of a vector field along a given path or curve \( C \). For a vector field \( \mathbf{F} \) and a curve \( C \), the line integral \( \int_C \mathbf{F} \cdot d\mathbf{r} \) can be thought of as adding up all the tiny projections of \( \mathbf{F} \) onto the tangent of the curve \( C \), integrated over the path.

• The curve is parametric, represented by \( \mathbf{r}(t) \), where \( t \) is the parameter.
• The notation \( d\mathbf{r} \) stands for the differential element of the curve, often reduced to the direction along the curve.

To calculate a line integral, you must trace the vector field through each point of the curve and compute the dot product of the field with the differential of the curve. In our problem, the curve \( C \) is defined by \( \mathbf{r}(t) = t\mathbf{i} + t^2\mathbf{j} + t^3\mathbf{k} \), and we found that integrating along this path leveraged the potential function derived from its gradient, making the computation of \( \int_C \mathbf{F} \cdot d\mathbf{r} \) significantly simpler.

The Fundamental Theorem of Line Integrals connects the concepts of scalar potential functions with line integrals in very insightful ways. It states that if a vector field \( \mathbf{F} \) is a conservative field, meaning \( \mathbf{F} = abla f \) for a function \( f \), then the line integral of \( \mathbf{F} \) over a curve \( C \) is equal to the difference in \( f \) evaluated at the endpoints of the curve. This remarkable result is expressed as:\[\int_C \mathbf{F} \cdot d\mathbf{r} = f(\mathbf{r}(b)) - f(\mathbf{r}(a))\]This theorem simplifies work significantly, especially when dealing with vector fields that are gradients of potential functions because:

• It eliminates the need to compute the line integral through parametrization or direct integration for conservative fields.
• One only needs to evaluate the potential function \( f \) at the endpoints of the curve \( C \).

In the given problem, we applied this theorem to find \( \int_C \mathbf{F} \cdot d\mathbf{r} \) as simply \( f(1, 1, 1) - f(0, 0, 0) \), resulting in the straightforward calculation of \( 3e \). By understanding this fundamental theorem, students can solve complex problems involving vector fields more efficiently.