Vector Calculus Summary

Line integrals

∫Cf(x,y)ds=∫abf(x(t),y(t))(dxdt)2+(dydt)2dt=∫abf(r(t))|r′(t)|dt\int^{\ }_{C}{f(x,y)}ds=\int^{b}_{a}{f\left(x(t),y(t)\right)\sqrt{\left(\frac{dx}{dt}\right)^{2}+\left(\frac{dy}{dt}\right)^{2}}\ dt}=\int^{b}_{a}{f\left(\mathbf{r}(t)\right)\left|\mathbf{r}'(t)\right|\ dt}

Or

∫Cf(x,y,z)ds=∫abf(x(t),y(t),z(t))(dxdt)2+(dydt)2+(dzdt)2dt=∫abf(r(t))|r′(t)|dt\int^{\ }_{C}{f(x,y,z)}ds=\int^{b}_{a}{f\left(x(t),y(t),z(t)\right)\sqrt{\left(\frac{dx}{dt}\right)^{2}+\left(\frac{dy}{dt}\right)^{2}+\left(\frac{dz}{dt}\right)^{2}}\ dt}=\int^{b}_{a}{f\left(\mathbf{r}(t)\right)\left|\mathbf{r}'(t)\right|\ dt}

∫CF⋅dr=∫CF(r(t))⋅r′(t)dt=∫CPdx+Qdy\int^{\ }_{C}{\mathbf{F}\cdot d\mathbf{r}}=\int^{\ }_{C}{\mathbf{F}\left(\mathbf{r}(t)\right)\cdot\mathbf{r}'(t)\ dt}=\int^{\ }_{C}{P\ dx+Q\ dy} OR Pdx+Qdy+Rdz{P\ dx+Q\ dy+R\ dz}

(These really mean∫CPdx+∫CQdy\int^{\ }_{C}{P\ dx+\int^{\ }_{C}{Q\ dy}} and ∫CPdx+∫CQdy+∫CRdz\int^{\ }_{C}{P\ dx+\int^{\ }_{C}{Q\ dy}}+\int^{\ }_{C}{R\ dz})

Note: We generally parameterize these.

Fundamental Theorem for Line Integrals

∫C∇f⋅dr=f(r(b))−f(r(a)),\int^{\ }_{C}{\mathbf{\nabla}f\cdot d\mathbf{r}}=f\left(\mathbf{r}(b)\right)-f\left(\mathbf{r}(a)\right),where r(t),a≤t≤b\mathbf{r}(t),\ \ \ a\leq t\leq b describes CC

=f(x2,y2)−f(x1,y1)=f\left(x_{2},\ y_{2}\right)-f\left(x_{1},\ y_{1}\right) or =f(x2,y2,z2)−f(x1,y1,z1)=f\left(x_{2},\ y_{2},\ z_{2}\right)-f\left(x_{1},\ y_{1},\ z_{1}\right)

∫CF⋅dr=0\int^{\ }_{C}{\mathbf{F}\cdot d\mathbf{r}}=0 for all closed paths CC ⇔\Longleftrightarrow∫CF⋅dr\int^{\ }_{C}{\mathbf{F}\cdot d\mathbf{r}} is independent of path ⇒\LongrightarrowFF is a conservative vector field ⇒∂Q∂x=∂P∂y\Longrightarrow\ \ \frac{\partial Q}{\partial x}=\frac{\partial P}{\partial y}

The last implication becomes if and only if and only if (⇔\Longleftrightarrow) if the partial derivatives are continuous throughout an open, simply connected region D, the domain of the vector field F\mathbf{F}.

If F\mathbf{F} is conservative, then F=∇f\mathbf{F}=\mathbf{\nabla}f for some potential function ff, and we can use the Fundamental Theorem for Line Integrals. If F\mathbf{F} is not conservative, we use the parameterized form given above(∫CF(r(t))⋅r′(t)dt)\left(\int^{\ }_{C}{\mathbf{F}\left(\mathbf{r}(t)\right)\cdot\mathbf{r}'(t)\ dt}\right), which becomes ∬D(∂Q∂x−∂P∂y)dA\iint^{\ }_{D}\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)dA if CC is the boundary of the closed region DD, by Green’s Theorem.

(Note: Green’s Theorem is stated below.)

To determine whether or not F\mathbf{F} is conservative (that is, whether or not to use the Fundamental Theorem for Line Integrals), in ℝ2\mathbb{R}^{2}, check if

∂P∂y=∂Q∂x\frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x}

In ℝ3\mathbb{R}^{3}, check if curlF=∇×F=𝟎\mbox{curl}\ \mathbf{F}=\mathbf{\nabla}\times\mathbf{F}=\mathbf{0}. If we find that F\mathbf{F} is conservative, we find the potential function ff by integrating:

f=∫Pdx,f=∫Qdy,(andinℝ3),f=∫Rdzf=\int^ {}_{}{P\ dx},\ \ f=\int^ {}_{}{Q\ dy},\ \ \ \ \ \ \ \left(\mbox{and}\ \mbox{in}\ \mathbb{R}^{3}\right),\ \ \ \ \ \ f=\int^ {}_{}{R\ dz}

Green’s Theorem

∫CF⋅dr=∮CPdx+Qdy=∫∂DPdx+Qdy=∬D(∂Q∂x−∂P∂y)dA\int^{\ }_{C}{\mathbf{F}\cdot d\mathbf{r}}=\oint^{\ }_{C}{P\ dx+Q\ dy}=\int^{\ }_{\partial D}{P\ dx+Q\ dy}=\iint^{\ }_{D}{\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)\ dA}

Notes: C=∂DC=\partial D is the simple, positively oriented boundary curve of DD. The symbol ∮C\oint^{\ }_{C} is used to indicate positive orientation.

Area of a parametric surface

A(S)=∬D|ru×rv|dA,whereuandvareparametersA(S)=\iint^{\ }_{D}{\left|\mathbf{r}_{u}\times\mathbf{r}_{v}\right|\ dA},\ \mbox{where}\ u\ \mbox{and}\ v\ \mbox{are}\ \mbox{parameters}

If xx and yy are the parameters, we have

A(S)=∬D1+(∂z∂x)2+(∂z∂y)2dAA(S)=\iint^{\ }_{D}{\sqrt{1+\left(\frac{\partial z}{\partial x}\right)^{2}+\left(\frac{\partial z}{\partial y}\right)^{2}}\ dA}

Surface integrals

∬Sf(x,y,z)dS=∬Df(r(u,v))|ru×rv|dA\iint^{\ }_{S}{f(x,\ y,\ z)\ dS=\iint^{\ }_{D}{f\left(\mathbf{r}(u,\ v)\right)}}\left|\mathbf{r}_{u}\times\mathbf{r}_{v}\right|\ dA

Note that dS=|ru×rv|dAdS=\left|\mathbf{r}_{u}\times\mathbf{r}_{v}\right|\ dA.

∬SF⋅dS=∬SF⋅ndS=∬DF⋅(ru×rv)dA\ \iint^{\ }_{S}{\mathbf{F}\cdot d\mathbf{S}}=\iint^{\ }_{S}{\mathbf{F}\cdot\mathbf{n}\ dS}=\iint^{\ }_{D}{\mathbf{F}\cdot\left(\mathbf{r}_{u}\times\mathbf{r}_{v}\right)dA}

Note: dS=ndS=|ru×rv|dAd\mathbf{S=n}\ dS=\left|\mathbf{r}_{u}\times\mathbf{r}_{v}\right|\ dA, where n\mathbf{n} is a unit normal vector to the surface SS and |ru×rv|\left|\mathbf{r}_{u}\times\mathbf{r}_{v}\right| is a normal vector to SS.

If xx and yy are the parameters, we have

∬SF⋅dS=∬D(−P∂g∂x−Q∂g∂y+R)dA,\iint^{\ }_{S}{\mathbf{F}\cdot d\mathbf{S}}=\iint^{\ }_{D}{\left(-P\frac{\partial g}{\partial x}-Q\frac{\partial g}{\partial y}+R\right)dA},\

for upward orientation. The signs of the integrand change for downward orientation.

Stokes’ Theorem

∫CF⋅dr=∬ScurlF⋅dS,\int^{\ }_{C}{\mathbf{F}\cdot d\mathbf{r}=\iint^{\ }_{S}{\mbox{curl}\ \mathbf{F}}\cdot d\mathbf{S}},

where CC is the positively oriented piecewise-smooth boundary curve of SS, an oriented piecewise-smooth surface.

The Divergence Theorem

∬SF⋅dS=∭EdivFdV,\iint^{\ }_{S}{\mathbf{F}\cdot d\mathbf{S}=\iiint^{\ }_{E}{\mbox{div}\ \mathbf{F}\ dV}},

where SS is the boundary surface of EE, a solid region whose surfaces are continuous, with outward orientation.