07 Vector Fields Free Online FlashCards

What is a vector field?

A vector field assigns a vector to every point in a region of space.

What is the standard 2D vector-field form?

In two dimensions, a vector field has the form $\mathbf{F}(x,y)=\langle P(x,y),Q(x,y)\rangle$.

How is a vector line integral computed parametrically?

For $\mathbf{r}(t)$, the line integral is $\int_C\mathbf{F}\cdot d\mathbf{r}=\int_a^b\mathbf{F}(\mathbf{r}(t))\cdot\mathbf{r}'(t)\,dt$.

When is a vector field conservative?

A field is conservative if $\mathbf{F}=\nabla f$ for some scalar potential function $f$.

What does path independence mean?

If $\mathbf{F}=\nabla f$, then $\int_C\mathbf{F}\cdot d\mathbf{r}=f(B)-f(A)$, so the value depends only on the endpoints.

What is the line integral around a closed curve in a conservative field?

The line integral around every closed curve is zero: $\oint_C\mathbf{F}\cdot d\mathbf{r}=0$.

What test identifies a conservative 2D field?

On a simply connected domain, continuous first partials and $P_y=Q_x$ imply that $\mathbf{F}=\langle P,Q\rangle$ is conservative.

What cross-partial conditions apply in 3D?

For $\mathbf{F}=\langle P,Q,R\rangle$, conservativeness requires $P_y=Q_x$, $P_z=R_x$, and $Q_z=R_y$.

How does a gradient relate to level sets?

Since $\mathbf{F}=\nabla f$, the field is perpendicular to the level curves or surfaces of $f$.

How is force related to potential energy?

If $U$ is potential energy, the force is $\mathbf{F}=-\nabla U$, so it points toward decreasing potential energy.

What differential equation describes a 2D streamline?

A 2D streamline for $\mathbf{v}=\langle P,Q\rangle$ satisfies $\frac{dy}{dx}=\frac{Q(x,y)}{P(x,y)}$, where $P\neq0$.

What is the divergence formula in 3D?

For $\mathbf{F}=\langle P,Q,R\rangle$, divergence is $\nabla\cdot\mathbf{F}=P_x+Q_y+R_z$.