Chapter 14: Vector Calculus

MATH2210

Divergence of F, where \overrightarrow{F}=\langle M,N,P\rangle

\nabla\bullet\overrightarrow{F}=\frac{\partial M}{\partial x}+\frac{\partial N}{\partial y}+\frac{\partial P}{\partial z}

Curl of F, where \overrightarrow{F}=\langle M,N,P\rangle

\nabla\times\overrightarrow{F}=\langle\frac{\partial P}{\partial y}-\frac{\partial N}{\partial z},\frac{\partial M}{\partial z}-\frac{\partial P}{\partial x},\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}\rangle

Definition of the line integral

\int_{C}f\left(x,y\right)ds=\lim_{\left\Vert P\right\Vert\to0}\sum_{i=1}^{n}f\left(\overline{x_{i}},\overline{y_{i}}\right)\Delta s_{i}

Line integral of two variables formula

\int_{C}f\left(x,y\right)ds=\int_{a}^{b}f\left(x\left(t\right),y\left(t\right)\right)\sqrt{\left\lbrack x^{\prime}\left(y\right)\right\rbrack^2+\left\lbrack y^{\prime}\left(t\right)\right\rbrack^2}dt

Line integral of three variables formula

\int_{C}f\left(x,y,z\right)ds=\int_{a}^{b}f\left(x\left(t\right),y\left(t\right),z\left(t\right)\right)\sqrt{\left\lbrack x^{\prime}\left(y\right)\right\rbrack^2+\left\lbrack y^{\prime}\left(t\right)\right\rbrack^2+\left\lbrack z^{\prime}\left(t\right)\right\rbrack}dt

Definition of work

W=\int_{C}\overrightarrow{F}\bullet\overrightarrow{T}ds=\int_{C}\overrightarrow{F}\bullet\overrightarrow{r^{\prime}}dt=\int_{C}\overrightarrow{F}\bullet d\overrightarrow{r}=\int_{C}Mdx+Ndy+Pdz

Fundamental theorem of line integrals

\int_{C}\nabla f\left(\overrightarrow{r}\right)\bullet d\overrightarrow{r}=f\left(\overrightarrow{b}\right)-f\left(\overrightarrow{a}\right)

Green’s Theorem

\iint_{S}\left(\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}\right)dA=\oint_{C}Mdx+Ndy

Vector forms of Green’s Theorem

• \oint_{C}\overrightarrow{F}\bullet\overrightarrow{n}ds=\iint_{S}\nabla\bullet\overrightarrow{F}dA=\Phi_{C} , where \overrightarrow{n}=\langle y^{\prime},-x^{\prime}\rangle

• \oint_{C}\overrightarrow{F}\bullet\overrightarrow{T}ds=\iint_{S}\left(\nabla\times\overrightarrow{F}\right)\bullet\overrightarrow{k}dA , where \overrightarrow{T}=\langle x^{\prime},y^{\prime}\rangle

Definition of the surface integral

\iint_{G}g\left(x,y,z\right)dS=\lim_{\left\Vert P\right\Vert\to0}\sum_{i=1}^{n}g\left(\overline{x_{i}},\overline{y_{i}},\overline{z_{i}}\right)\Delta S_{i}

Surface integral formula

\iint_{G}g\left(x,y,z\right)dS=\iint_{R}g\left(x,y,f\left(x,y\right)\right)\sqrt{f_{x}^2+f_{y}^2+1}dydx

Flux formula

\Phi_{G}=\iint_{G}\overrightarrow{F}\bullet\overrightarrow{n}dS=\iint_{R}\left(-Mf_{x}-Nf_{y}+P\right)dxdy

Gauss’s Divergence Theorem

\iint_{\partial S}\overrightarrow{F}\bullet\overrightarrow{n}dS=\iiint_{S}\nabla\bullet\overrightarrow{F}dV=\iiint_{S}\left(\frac{\partial M}{\partial x}+\frac{\partial N}{\partial y}+\frac{\partial P}{\partial z}\right)dV

Stokes’s Theorem

\oint_{\partial S}\overrightarrow{F}\bullet\overrightarrow{T}ds=\iint_{S}\left(\nabla\times\overrightarrow{F}\right)\bullet\overrightarrow{n}dS