Vector Fields
Table of Contents
A vector field is something of the form:
\[ \vec{F} = M\hat{i} + N\hat{j} \]
where \(M\) and \(N\) are functions of \(x\) and \(y\). Concretely, this means that for each point in the plane \((x,y)\), you have a vector \(\vec{F}\) that depends on \((x,y)\): hence, a vector field or a field of vectors.
1. Gradient Field
The gradient of a function, \(\nabla f\), can also be considered to be a vector field. In physics, the gradient field is directly related to the concept of potential.
To test if a vector field is a gradient field, we can consider that if \(\vec{F} = \nabla f\), \(\vec{F} = \langle M, N \rangle\) where \(M=f_x\) and \(N=f_y\). Since we know that \(f_{xy} = f_{yx}\), then it must be true that:
\begin{align} M_y = N_x \end{align}The converse is only true if the vector field is defined everywhere, or in a simply-connected region, and in those cases we can use this property to show that we have a gradient field. This is summarized with curl.
2. Path Independence
The gradient field is a good example of a field that exhibits the property of path independence. Path independence means that the line integral of a curve with this field is independent of the path between the endpoints.
In other words, if we have two curves \(C_1\) and \(C_2\) that have the same endpoints, then:
\[ \int_{C_1} \vec{F} \cdot \text{ d}\vec{r} = \int_{C_2} \vec{F} \cdot \text{ d}\vec{r} \]
This is due to the result obtained from the fundamental theorem of calculus for line integrals.
3. Conservative Fields
The gradient field is also a good example of a conservative field. A conservative field is defined as a field such that the line integral of the field on a closed loop \(C\) is equal to zero. This directly follows from the concept of path independence:
\[ \int_C \vec{F} \cdot \text{ d}\vec{r} = f(P) - f(P) = 0 \]