08 Line Integrals Free Online FlashCards

How is a parametric curve represented?

A parametric curve is represented by $\mathbf r(t)=\langle x(t),y(t)\rangle$ or $\mathbf r(t)=\langle x(t),y(t),z(t)\rangle$ over an interval $a\le t\le b$.

What is the arc-length differential?

The arc-length differential is $ds=\lVert\mathbf r'(t)\rVert\,dt$.

What is the formula for a scalar line integral?

A scalar line integral is $\int_C f\,ds=\int_a^b f(\mathbf r(t))\lVert\mathbf r'(t)\rVert\,dt$.

How is the mass of a wire expressed as a line integral?

For a wire with linear density $\rho$, its mass is $m=\int_C\rho\,ds$. If density is constant, $m=\rho L$.

What is the parametrized formula for a vector line integral?

A vector line integral is $\int_C\mathbf F\cdot d\mathbf r=\int_a^b\mathbf F(\mathbf r(t))\cdot\mathbf r'(t)\,dt$.

How is work by a force field calculated along a curve?

The work is $W=\int_C\mathbf F\cdot d\mathbf r$. A force parallel to motion contributes positive work; an opposing force contributes negative work.

What does the Fundamental Theorem for Line Integrals state?

If $\mathbf F=\nabla f$, then $\int_C\mathbf F\cdot d\mathbf r=f(\mathbf r(b))-f(\mathbf r(a))$; only the endpoints matter.

When is a vector field conservative?

A vector field is conservative if there is a scalar potential $f$ such that $\mathbf F=\nabla f$.

What is the tangent vector of $\mathbf r(t)$?

The tangent vector is $\mathbf r'(t)$, obtained by differentiating each component of the parametrization with respect to $t$.

How is the length of a smooth curve calculated?

The length is $L=\int_a^b\lVert\mathbf r'(t)\rVert\,dt$.

Does reversing orientation change a scalar line integral?

No. Reversing the curve leaves a scalar line integral unchanged because $ds$ represents positive length.

How is a planar vector line integral written in differential form?

In the plane, $\int_C\mathbf F\cdot d\mathbf r=\int_C P\,dx+Q\,dy$ for $\mathbf F=\langle P,Q\rangle$.