09 Green’s Theorem and Planar Vector Calculus Free Online FlashCards

What does a vector line integral measure?

The line integral $\int_C \mathbf F\cdot d\mathbf r=\int_C P\,dx+Q\,dy$ measures the work done by the field along the path.

What does positive orientation mean in the plane?

For a positively oriented closed curve, the region stays on the left; in the plane, this normally means counterclockwise traversal.

What is the planar curl formula?

For $\mathbf F=\langle P,Q\rangle$, the scalar planar curl is $\operatorname{curl}\mathbf F=\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}$.

What is the planar divergence formula?

For $\mathbf F=\langle P,Q\rangle$, the divergence is $\operatorname{div}\mathbf F=\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}$.

What is Green’s theorem in circulation form?

Green’s circulation theorem states $\oint_C P\,dx+Q\,dy=\iint_D\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)dA$.

What is Green’s theorem in flux form?

Green’s flux theorem states $\oint_C \mathbf F\cdot\mathbf n\,ds=\iint_D\left(\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}\right)dA$.

How is outward flux written as a line integral?

For counterclockwise orientation, the outward-normal element is $\mathbf n\,ds=\langle dy,-dx\rangle$, so $\mathbf F\cdot\mathbf n\,ds=P\,dy-Q\,dx$.

What are curl and divergence for $\mathbf F=\langle -y,x\rangle$?

For $\mathbf F=\langle -y,x\rangle$, the curl is $2$ and the divergence is $0$, so the field rotates without net source or sink behavior.

How does a potential function determine a line integral?

A conservative field has a potential $\phi$ with $\mathbf F=\nabla\phi$; its line integral equals $\phi(\text{endpoint})-\phi(\text{starting point})$.

When does zero curl imply a conservative field?

On a suitable simply connected region, zero curl implies that a continuously differentiable field is conservative.

Why is the punctured-plane field not conservative?

The field $\left\langle\frac{-y}{x^2+y^2},\frac{x}{x^2+y^2}\right\rangle$ has zero curl on its domain but circulation $2\pi$ around the unit circle, so it is not conservative there.

How are boundaries oriented around holes?

For a region with holes, orient the outer boundary counterclockwise and every inner boundary clockwise.