Chapter 4

Previous: Chapter 3

Vector Fields

Let \(D\) be the region in the 2D space \(R^2\). A vector field on \(D\) is a vector \(\textbf{F}\) of the form:
\(F(x,y) = P(x,y)\textbf{i} + Q(x,y)\textbf{j}\)

Vector Fields in 3D

A field vector on \(D\) is a vector \(\textbf{F}\) of the form:
\(F(x,y,z) = P(x,y,z)\textbf{i} + Q(x,y,z)\textbf{j} + R(x,y,z) \textbf{k}\)

Another example of a vector field is the gradient field (or gradient vector) denoted by \(\nabla f = \frac{\partial f}{\partial x} \textbf{i} + \frac{\partial f}{\partial y} \textbf{j}\)
Respectively: \(\nabla f = \frac{\partial f}{\partial x} \textbf{i} + \frac{\partial f}{\partial y} \textbf{j} + \frac{\partial f}{\partial z} \textbf{k}\)

Line integrals of Vector fields

Given a curve \(C: r(t), a\leq r \leq b\) and a vector field \(\textbf{F}\) representing a force, both in the same space, we would like to calculate the total work done by \(\textbf{F}\) a particle along the curve \(C\) from the point \(t=a\) (initial point) to the point \(t=b\) (terminal point).
![[Pasted image 20240917182454.png]]
work done, called the line integral of \(F\) along \(C\) and denoted by:
\(\int_{C}\textbf{F}\cdot d\textbf{r} = \int_{a}^{b} \textbf{F}(r(t)) \cdot \textbf{r}\prime(t) dt = \int_{a}^{b}P(x(t),y(t)) \cdot x \prime (t) +Q(x(t), y(t)) y \prime (t)dt\)

\(\int_{C}\textbf{F}\cdot d\textbf{r} = \int_{a}^{b} \textbf{F}(r(t)) \cdot \textbf{r}\prime(t) dt = \int_{a}^{b}P(x(t),y(t),z(t)) \cdot x \prime (t) +Q(x(t), y(t),z(t)) \cdot y \prime (t) + R(x(t), y(t), z(t)) \cdot z \prime (t)dt\)

Conservative fields

A vector field \(\textbf{F}\) is said to be conservative, if there is a scalar function \(f\) such that \(\textbf{F} = \nabla f\)
where \(\nabla f = \frac{\partial f}{\partial x} \textbf{i} + \frac{\partial f}{\partial y} \textbf{j}\)
\(\nabla f = \frac{\partial f}{\partial x} \textbf{i} + \frac{\partial f}{\partial y} \textbf{j} +\frac{\partial f}{\partial z} \textbf{k}\)

Line Integrals of Conservative fields

Note:

The line integral of F along a smooth curve \(C: \textbf{r}(t), a \leq t \leq b\) joining the point \(\textbf{r}(a)\) to the point \(\textbf{r}(b)\), the work done by F in moving a particle along \(C\) from \(t=a\) to \(t=b\)
In other words
\(\int_{c}\textbf{F} \cdot d\textbf{r}=\left(\int^{b}_{a}\textbf{F}(r(t)) \cdot \textbf{r} \prime (t) dt\right)=f(\textbf{r}(b)) - f(\textbf{r}(a))\)
When calculating the line integral of a conservative field over any curve, no integration is required.
That means when the curve is closed,
\(f(\textbf{r}(b)) = f(\textbf{r}(a))\)
\(f(\textbf{r}(b)) - f(\textbf{r}(a)) = 0\)

When a curve \(C\) is closed it is customary to write\(\int_{c}\textbf{F} \cdot d\textbf{r}\) as \(\oint_{c}\textbf{F} \cdot d\textbf{r}\)

Test for conservative fields

\(\textbf{F} (x,y, z)\) is conservative if and only if
\(P_{x}= Q_y\) and \(Q_{z}= R_{y}\) and \(P_{z} =R_{x}\)

Green's Theorem in 2D

\(\int_{c}\textbf{F}\cdot d\textbf{r}\) can be written as \(\int_{a}^{b}P(x(t),y(t)) \cdot x \prime (t) +Q(x(t), y(t)) y \prime (t)dt\)
It is customary to write
\(\int_{a}^{b}P(x(t),y(t)) \cdot x \prime (t) dt = \int_{c} P dx\)
\(\int_{a}^{b}Q(x(t),y(t)) \cdot y \prime (t) dt = \int_{c} Q dy\)
\(\int_{c} P \space dx + Q \space dy\)

A simple closed curve is one that does not cross itself. The positive orientation of a simple closed curve refers to a single counterclockwise traversal of the curve.

If \(D\) is the region enclosed by a simple, closed and positively oriented curve \(C\), then \(\oint_{c}\textbf{F} \cdot d\textbf{r} =\oint_{c} P \space dx + Q \space dy = \iint_{D}(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y})\) dA
When \(C\) is negatively oriented the sign will be negative.

When closed curve \(C\) is made up of two or more curves
\(\oint_{c} P \space dx + Q \space dy = \oint_{c_1} P \space dx + Q \space dy + \oint_{c_2} P \space dx + Q \space dy \space ... \space \oint_{c_n} P \space dx + Q \space dy\)

Curl and Divergence

The Curve Vector

The curl vector of \(\textbf{F}\) denoted by curl \(\textbf{F}\) is a vector defined by
\((\frac{\partial Q}{\partial z} - \frac{\partial R}{\partial y})\textbf{i} + (\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x})\textbf{j} + (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y})\textbf{k}\)

It can also be calculated by the cross product of the operator and the vector field:
\(\nabla \times \textbf{F} = \pmatrix{\frac{\partial }{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z}} \times \pmatrix{P\\ Q \\ R}=\pmatrix{\frac{\partial Q}{\partial z} - \frac{\partial R}{\partial y} \\ \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \\ \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}}\)
\(\nabla = \frac{\partial }{\partial x}\textbf{i} + \frac{\partial}{\partial y} \textbf{j} + \frac{\partial}{\partial z}\textbf{k}\)

The Divergence

The divergence of \(\textbf{F}\), denoted by div \(\textbf{F}\), is a scalar defined by
\(div \space \textbf{F} = \frac{\partial P}{\partial x}\textbf{i} + \frac{\partial Q}{\partial y} \textbf{j} + \frac{\partial R}{\partial z}\textbf{k}\)
We can also write divergence of \(\textbf{F}\) as
\(div \space \textbf{F} = \nabla \cdot \textbf{F}\)

Next: Chapter 5