Definition: Vector fields in $\mathbb{R}^2$
A vector field in $\displaystyle {\mathbb{R}}^2$ is an assignment of a two dimensional vector $\vec{F}(x,y)$ to each point $(x,y)\in{\mathcal{D}}\subseteq{\mathbb{R}}^2$
$$ \vec{F}(x,y) =\left(\begin{array}{r}P(x,y)\\Q(x,y)\end{array}\right) =P(x,y)\vec{i}+Q(x,y)\vec{j} $$Definition: Vector fields in $\mathbb{R}^3$
A vector field in $\displaystyle {\mathbb{R}}^3$ is an assignment of a three dimensional vector $\vec{F}(x,y,z)$ to each point $(x,y,z)\in{\mathcal{D}}\subseteq{\mathbb{R}}^3$
$$ \vec{F}(x,y,z) = \left(\begin{array}{r} P(x,y,z) \\Q(x,y,z) \\R(x,y,z) \end{array}\right) = P(x,y,z)\vec{i}+Q(x,y,z)\vec{j}+R(x,y,z)\vec{k} $$Definition: Unit vector fields
A vector field $\vec{F}$ is unit vector field if the magnitude of each vector is one.
Definition: Gradient vector fields
A vector field $\vec{F}$ is gradient or conservative if there is a scalar function $f$ such that $\displaystyle \nabla f = \vec{F}$. The function $f$ is called potential function.
Theorem: Uniqueness of potential functions
Let $\vec{F}$ be a conservative vector field on an open connected domain. Let $f$ and $g$ be continuous functions such that
$$\displaystyle \nabla f = \vec{F}$$and
$$\displaystyle \nabla g = \vec{F}$$then there is a constant $c$ such that
$$f=g+c$$Theorem: Cross partial property
Let $\displaystyle \vec{F}(x,y,z) = \left(\begin{array}{r} P(x,y,z) \\Q(x,y,z) \\R(x,y,z) \end{array}\right) $ be a vector field on an open connected domain where $\vec{F}$ has continuous first partial derivatives. Then $\vec{F}$ is conservative if and only if
$$ \displaystyle \frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x} \qquad \frac{\partial P}{\partial z} = \frac{\partial R}{\partial x} \qquad \frac{\partial Q}{\partial z} = \frac{\partial R}{\partial y} $$Definition: Scalar integral
Let $f$ be a smooth function whose domain include the smooth curve $\mathbf{C}$ parametrized by
$$\vec{r}(t) = x(t)\vec{i} +y(t)\vec{j} +z(t)\vec{k},\qquad a\le t\le b $$The scalar integral of $f$ along $\mathbf{C}$ is
$$ \int_{\mathbf{C}}f\,\mathrm{d}\,{s} =\lim_{n\to\infty}\sum_{i=1}^n f(x_i^*)\Delta s $$assuming the limit exists.
Theorem: Evaluating scalar integrals
Let $f$ be a smooth function whose domain include the smooth curve $\mathbf{C}$ parametrized by
$$\vec{r}(t) = x(t)\vec{i} +y(t)\vec{j} +z(t)\vec{k},\qquad a\le t\le b $$Then
$$ \begin{array}{rcl}\displaystyle \int_{\mathbf{C}}f\,\mathrm{d}\,{s} &=&\int_{a}^{b} f\left({\vec{r}\left(t\right)}\right) \|\vec{r}\,'\left(t\right)\|\,\mathrm{d}\,{t} \\ &=&\int_{a}^{b} f\left({\vec{r}\left(t\right)}\right) \sqrt{ {\left(x'(t)\right)}^2 +{\left(y'(t)\right)}^2 +{\left(z'(t)\right)}^2 } \,\mathrm{d}\,{t} \end{array} $$Definition: Vector Line integral
Let $ \vec{F}(x,y,z) $ be a vector field and $\mathbf{C}$ a curve. The vector line integral of $\vec{F}$ along $\mathbf{C}$ is
$$ \int_{\mathbf{C}}\vec{F}\cdot \,\mathrm{d}\,{\vec{r}} =\int_{\mathbf{C}}\vec{F}\cdot \vec{T}\,\mathrm{d}\,{s} =\lim_{n\to\infty}\sum_{i=1}^n \vec{F}(P_i^*)\cdot\vec{T}(P_i^*)\Delta s $$assuming the limit exists.
Definition: Flux
Let $ \vec{F}(x,y) = P(x,y)\vec{i}+ Q(x,y)\vec{j}$ be a vector field and $\mathbf{C}$ a curve. The flux of $\vec{F}$ across $\mathbf{C}$ is
$$ \int_{\mathbf{C}}\vec{F}\cdot\vec{N} \,\mathrm{d}\,{s} =\int_{\mathbf{C}}\vec{F}\cdot\frac{\vec{n}}{\|\vec{n}\|} \,\mathrm{d}\,{s} $$Theorem: Evaluating scalar integrals
If $\mathbf{C}=\vec{r}(t)=x(t)\vec{i}+y(t)\vec{j}$ where $a\le t\le b$.
$$ \int_{\mathbf{C}}\vec{F}\cdot\vec{N} \,\mathrm{d}\,{s} =\int_{a}^{b}\vec{F}\left(\vec{r}(t)\right) \cdot\vec{n} \,\mathrm{d}\,{t} $$Definition: Circulation
The line integral of a vector field $ \vec{F} $ along a simple closed curve $\mathbf{C}$ is called circulation of $\vec{F}$ along $\mathbf{C}$.
$$\oint_{\mathbf{C}}\vec{F}\cdot\vec{T}\,\mathrm{d}\,{s}$$A curve $\mathbf{C}=\vec{r}(t)$ for $a\le t\le b$ is closed if
$$\vec{r}(a)=\vec{r}(b)$$A curve $\mathbf{C}=\vec{r}(t)$ for $a\le t\le b$ is simple if
$$\vec{r}(t)$$is one to one on $t\in(a,b)$.
A region $D$ is connected if for any two points $P$ and $Q$ in $D$ there is a (continuous) path that starts at $P$ and ends at $Q$ that has trace contained entirely in $D$.
A region $D$ is simply connected if for any simple closed curve that lies entirely in $D$ the curve can be continuously shrunk to a point while staying entirely in $D$.
Theorem: Fundamental Theorem for line integrals
Let $\mathbf{C}=\vec{r}(t)$ for $a\le t\le b$ be a smooth curve and $f$ has continuous partial derivatives.
$$ \int_{\mathbf{C}} \nabla f \cdot \,\mathrm{d}\,{\vec{r}} = f\left(\vec{r}\left(b\right)\right) - f\left(\vec{r}\left(a\right)\right) $$Definition: Path independent
Let $\vec{F}$ be a vector field with domain ${\mathcal{D}}$. Then $\vec{F}$ is path independent or independent of path if
$$ \int_{\mathbf{C}_1} \vec{F}\cdot \,\mathrm{d}\,{\vec{r}} =\int_{\mathbf{C}_2} \vec{F}\cdot \,\mathrm{d}\,{\vec{r}} $$for any two paths $\mathbf{C}_1$ and $\mathbf{C}_2$ in ${\mathcal{D}}$ that have the same initial and terminal point.
Let $\displaystyle \vec{F}(x,y,z) = P(x,y,z)\vec{i} +Q(x,y,z)\vec{j}+R(x,y,z)\vec{k}$
| Evaluate | $\displaystyle g(x,y,z) + h(y,z) =\left(\int P(x,y,z)\,\mathrm{d}\,{x}\right)$ |
| Find $h_y(y,z)$ using | $ g_y(x,y,z)+ h_y(y,z) = Q(x,y,z)$ |
| Evaluate | $ v(y,z) + u(z) = \int h_y(y,z)\,\mathrm{d}\,{y}$ |
| Find $u_z(z)$ using | $ g_z(x,y,z)+v_z(y,z)+ u_z(z) = R(x,y,z)$ |
| Evaluate | $ u(z) = \int u_z(z)\,\mathrm{d}\,{z}$ |
Theorem: Green's Theorem in circulation form
Let ${\mathcal{D}}$ be an open connected regions bounded by a piece-wise smooth curve $\mathbf{C}=\vec{r}(t)$. Let $\vec{F}(x,y) = P(x,y)\vec{i}+Q(x,y)\vec{j}$ be a vector field with piecewise continuous partial derivatives then
$$ \oint_{\mathbf{C}}\vec{F}\cdot\,\mathrm{d}\,{\vec{r}} = \oint_{\mathbf{C}}P \,\mathrm{d}\,{x} +Q\,\mathrm{d}\,{y} = \iint_{{\mathcal{D}}}\left(Q_x - P_y\right)\,\mathrm{d}\,{A} $$Theorem: Green's Theorem in flux form
Let ${\mathcal{D}}$ be an open connected regions bounded by a piece-wise smooth curve $\mathbf{C}=\vec{r}(t)$. Let $\vec{F}(x,y) = P(x,y)\vec{i}+Q(x,y)\vec{j}$ be a vector field with piecewise continuous partial derivatives then
$$ \oint_{\mathbf{C}}\vec{F}\cdot\vec{N}\,\mathrm{d}\,{s} = \iint_{{\mathcal{D}}}\left(P_x + Q_y\right)\,\mathrm{d}\,{A} $$Let $$\vec{F}(x,y,z) = P(x,y,z)\vec{i}+Q(x,y,z)\vec{j}+R(x,y,z)\vec{k}$$ be a vector field for which $P_x$, $Q_y$ and $R_z$ exist
$$ \mathrm{div} \vec{F} =P_x +Q_y+R_z =\frac{\partial}{\partial\,x}\,P +\frac{\partial}{\partial\,y}\,Q +\frac{\partial}{\partial\,z}\,R $$$\vec{F}(x,y,z) = P(x,y,z)\vec{i}+Q(x,y,z)\vec{j}+R(x,y,z)\vec{k}$
$$ \begin{array}{rcl}\displaystyle \mathrm{curl} \vec{F} &=& \left(R_y-Q_z\right) \vec{i} +\left(P_z-R_x\right) \vec{j} +\left(Q_x-P_y\right) \vec{k} \\ &=& \left(\frac{\partial\,R}{\partial\,y} -\frac{\partial\,Q}{\partial\,z}\right)\vec{i} +\left(\frac{\partial\,P}{\partial\,z} -\frac{\partial\,R}{\partial\,x}\right)\vec{j} +\left(\frac{\partial\,Q}{\partial\,x} -\frac{\partial\,P}{\partial\,y}\right)\vec{k} \end{array} $$$\vec{F}(x,y,z) = P(x,y,z)\vec{i}+Q(x,y,z)\vec{j}+R(x,y,z)\vec{k}$
$$ \mathrm{curl} \vec{F} = \nabla\times\vec{F} =\left|\begin{array}{ccc} \vec{i} & \vec{j} & \vec{k} \\ \frac{\partial}{\partial\,x} &\frac{\partial}{\partial\,y} &\frac{\partial}{\partial\,z} \\ P & Q & R \end{array}\right| $$Theorem: Properties
Let $\vec{F}(x,y,z) = P(x,y,z)\vec{i}+Q(x,y,z)\vec{j}+R(x,y,z)\vec{k}$ has continuous second order partial derivatives then
$$ \mathrm{div} \left(\mathrm{curl} \left(\vec{F} \right)\right) = \nabla\cdot\left(\nabla\times\vec{F}\right) = 0 $$Theorem: Properties
Let $\vec{F}(x,y,z) = P(x,y,z)\vec{i}+Q(x,y,z)\vec{j}+R(x,y,z)\vec{k}$ be defined in space over a simply connected domain. If
$$ \mathrm{curl}\, \vec{F} = \vec{0} $$then $\vec{F}$ is conservative.
Theorem: Properties
if $\vec{F}(x,y,z) = P(x,y,z)\vec{i}+Q(x,y,z)\vec{j}+R(x,y,z)\vec{k}$ is conservative then
$$ \mathrm{curl}\, \vec{F} = \vec{0} $$A parametrized (parametric) surface is given by
$$\vec{\sigma}(u,v) = x(u,v)\vec{i} +y(u,v)\vec{j} +z(u,v)\vec{k} =\left(\begin{array}{c} x(u,v) \\y(u,v) \\z(u,v) \end{array}\right) $$The parameter domain of the surface is the set of values ${\mathcal{D}}$ in the $uv$-plane for which $\vec{\sigma}(u,v)$ is defined.
Parametrization
$$\vec{\sigma}(u,v) = x(u,v)\vec{i} +y(u,v)\vec{j} +z(u,v)\vec{k} =\left(\begin{array}{c} x(u,v) \\y(u,v) \\z(u,v) \end{array}\right) $$is smooth if
$$\vec{\sigma}_u(u,v)\times\vec{\sigma}_v(u,v)\ne \vec{0}$$and for any $u,v$ and $\vec{\sigma}_u$ and $\vec{\sigma}_v$ are continuous in the interior of the domain of $\vec{\sigma}$.
Let
$$\vec{\sigma}(u,v) = x(u,v)\vec{i} +y(u,v)\vec{j} +z(u,v)\vec{k} $$be smooth parametrization of a surface $\mathbf{S}$ such that $\vec{\sigma}(u,v)$ is one-to-one (i.e. $\mathbf{S}$ is traced only once) in its domain ${\mathcal{D}}$ then the surface area of $\mathbf{S}$ is
$$ A(\mathbf{S}) = \iint_{{\mathcal{D}}}\|\vec{\sigma}_u\times\vec{\sigma}_v\|\,\mathrm{d}\,{A} $$The surface integral of scalar valued function $f(x,y,z)$ over the piece-wise smooth surface $\mathbf{S}=\vec{r}(u,v)$ is
$$\iint_{\mathbf{S}}f(x,y,z)\,\mathrm{d}\,{\sigma} = \iint_{{\mathcal{D}}}f(r(u,v))\|\vec{r}_u\times\vec{r}_v\|\,\mathrm{d}\,{u}\,\mathrm{d}\,{v} $$A surface $\mathbf{S}\subset{\mathbb{R}}^3$ is orientable if there is a continuous vector field $\vec{N}$ such that $\vec{N}$ is non-zero and normal to $\mathbf{S}$ at each point of $\mathbf{S}$.
The surface integral of vector valued function $\vec{F}(x,y,z)$ over the piece-wise smooth surface $\mathbf{S}=\vec{r}(u,v)$ with unit normal $\vec{N}$ is
$$\iint_{\mathbf{S}}\vec{F}\,\mathrm{d}\,{\vec{\sigma}} = \iint_{{\mathcal{D}}}\vec{F}\cdot\vec{N}\,\mathrm{d}\,{\sigma} $$Let $\mathbf{S}$ be an orientable surface in ${\mathbb{R}}^3$ whose bounder is simple closed curve $\mathbf{C}$. Let $\vec{F}$ be smooth vector field defined on subspace of ${\mathbb{R}}^3$ that contains $\mathbf{S}$. Then
$$ \oint_{C}\vec{F}\cdot\,\mathrm{d}\,{\vec{r}} = \iint_{\mathbf{S}}\mathrm{curl}\, \vec{F} \cdot \vec{n}\,\mathrm{d}\,{\vec{r}} = \iint_{\mathbf{S}}\left(\nabla \times \vec{F} \right)\cdot \vec{n}\,\mathrm{d}\,{\vec{r}} $$Let $\Sigma$ be closed surface in ${\mathbb{R}}^3$ which bounds a volume $\mathbf{V}$. Let $\vec{F}$ be smooth vector field defined on subspace of ${\mathbb{R}}^3$ that contains $\Sigma$. Then
$$ \iint_{\Sigma}\vec{F}\cdot\,\mathrm{d}\,{\vec{\sigma}} = \iiint_{\mathbf{V}}\mathrm{div}\,\vec{F} \,\mathrm{d}\,{V} = \iiint_{\mathbf{V}}\left(\nabla \cdot\vec{F} \right)\,\mathrm{d}\,{V} $$