• Multivariate Calculus
• multivariate function
  • level curve (contour curve)/ level space
  • limit of multivariate function
  • continuity of multivariate function
  • partial derivative
    • higher partial derivative
      • Clairaut’s theorem
  • tangent plane
    • line approximation
  • increment
    • differentiable function
    • differential
  • implicit function
    • implicit function theorem
  • gradient
    • directional derivative
    • level surface
      • tangent plane of level surface
  • extrema
    • extrema subject to constraint
  • surface area
  • Jacobian
  • line integral
    • work
    • fundamental theorem of line integral
    • conservative vector field
      • independence of path of line integral
  • Green’s theorem
• nabla
  • Laplace operator
  • curl
  • divergence
  • Green’s theorem in vector form
• parametric surface
  • oriented surface
    • flux
  • Stoke’s theorem
  • divergence theorem (Gauss’s theorem)

Multivariate Calculus

multivariate function

$$\{f(x_1,x_2,\cdots )|(x_1,x_2,\cdots )\in D\}$$

level curve (contour curve)/ level space

curve/ space with equation

$$f(x_1,x_2,\cdots )=k$$

• $k$ is constant

limit of multivariate function

$$\underset{(x_1,x_2,\cdots )\to (a_1,a_2,\cdots )}{\lim }f(x_1,x_2,\cdots )=L$$

$\iff f(x_1,x_2,\cdots )$ approach $L$ as $(x_1,x_2,\cdots )$ approach $(a_1,a_2,\cdots )$ from any path within domain

• proof for the opposite using counterexample:
  point out that the function approach different value from two different direction
• the squeeze theorem hold

continuity of multivariate function

$$\underset{(x_1,x_2,\cdots )\to (a_1,a_2,\cdots )}{\lim }f(x_1,x_2,\cdots )=f(a_1,a_2,\cdots )$$

$\iff f$ continuous

partial derivative

$$\frac{\partial f}{\partial x_n}{|}_{x_1=a_1,\cdots ,x_n=a_n,\cdots }=f_{x_n}(a_1,a_2,\cdots )=g^{\prime }(a_n)$$

where

$$g(x_n)=f(a_1,\cdots ,x_n,a_{n+1},\cdots )$$

higher partial derivative

$$(f_{x_m}{)}_{x_n}=f_{x_m{x}_n}$$

Clairaut’s theorem

$f$ defined on $D$ containing $(a,b)$, $f_{xy},f_{yx}$ continuous

$$\Rightarrow f_{xy}(a,b)=f_{yx}(a,b)$$

tangent plane

$$z-z_0=f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)$$

line approximation

$$f(x,y)\approx f_x(a,b)(x-a)+f_y(a,b)(y-b)+f(a,b)$$

increment

$$\Delta z=f(a+\Delta x,b+\Delta y)-f(a,b)$$

differentiable function

$$\begin{gathered} \underset{\Delta x\to 0,\Delta y\to 0}{\lim }\Delta z=\underset{\Delta x\to 0,\Delta y\to 0}{\lim }\left(\frac{\partial z}{\partial x}+{\epsilon }_1\right)\Delta x+\left(\frac{\partial z}{\partial y}+{\epsilon }_2\right)\Delta y \\ =\underset{\Delta x\to 0,\Delta y\to 0}{\lim }\frac{\partial z}{\partial x}\Delta x+\frac{\partial z}{\partial y}\Delta y \end{gathered}$$

differential

for differentiable function

$$dz=\frac{\partial z}{\partial x}dx+\frac{\partial z}{\partial y}dy$$

implicit function

$$F(x,y)=0$$

implicit function theorem

given

1. $F$ differentiable by $x$ and $y$
2. $\frac{\partial F}{\partial y}\ne 0$

take derivative of both side

$$\frac{\partial F}{\partial x}\frac{dx}{dx}+\frac{\partial F}{\partial y}\frac{dy}{dx}=0\Rightarrow \frac{dy}{dx}=-\frac{\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial y}}$$

gradient

$$\nabla f(x_1,\cdots ,x_n)=⟨f_{x_1},\cdots ,f_{x_n}⟩$$

• product rule, $\nabla (fg)=(\nabla f)g+f(\nabla g)$

directional derivative

$\vec{u}$ is unit vector

$$\begin{gathered} D_{\vec{u}}=\underset{h\to 0}{\lim }\frac{f(x_1+u_{1}h,\cdots ,x_n+u_{n}h)-f(x_1,\cdots ,x_n)}{h} \\ =\nabla f(x_1,\cdots ,x_n)\cdot \vec{u} \end{gathered}$$

$\max (D_{\vec{u}}f)=|\nabla f|$ only when $\vec{u},\nabla f(x_1,\cdots ,x_n)$ have the same direction

level surface

level surface $F(x,y,z)=k$

line curve $\vec{r}(t)$ on the surface of the level surface

$$\nabla F\cdot {\vec{r}}^{\prime }=0$$

$\Rightarrow \nabla F$ is normal vector of tangent plane

• for level curve, $\nabla F$ is perpendicular to the curve

tangent plane of level surface

$$F_x(x-x_0)+F_y(y-y_0)+F_z(z-z_0)=0$$

extrema

point $(a,b)$

$$\{\begin{array}{l}{f}_x(a,b)=0\\ f_y(a,b)=0\end{array}$$

$\Rightarrow$ maximum or minimum or saddle point

$$D(a,b)=f_{xx}(a,b)f_{yy}(a,b)-f_{xy}^2(a,b)$$

• $D>0\Rightarrow$
  • $f_{xx}>0\Rightarrow$ local minimum
  • $f_{xx}<0\Rightarrow$ local maximum
• $D<0\Rightarrow$ saddle point

extrema subject to constraint

constraint $g(x,y,z)=k$

$$\nabla f\parallel \nabla g\Rightarrow \nabla f(x,y,z)=\lambda \nabla g(x,y,z)$$

• Lagrange multiplier $\lambda$

for additional constraint $h(x,y,z)=c$

two constraint’s intersection curve $C$

$$\nabla f,\nabla g,\nabla h\perp C\Rightarrow \nabla f(x,y,z)=\lambda \nabla g(x,y,z)+\mu \nabla h(x,y,z)$$

surface area

$$A(S)={\iint }_D\sqrt{1+f_x^2+f_y^2}dA$$

Jacobian

$x=x(u,v),y=y(u,v)$

$$\begin{gathered} \frac{\partial (x,y)}{\partial (u,v)}=\left|\begin{array}{cc}\frac{\partial x}{\partial u}& \frac{\partial x}{\partial v}\\ \frac{\partial y}{\partial u}& \frac{\partial y}{\partial v}\end{array}\right|=\frac{\partial x}{\partial u}\frac{\partial y}{\partial v}-\frac{\partial x}{\partial v}\frac{\partial y}{\partial u} \\ {\iint }_{R}f(x,y)dxdy={\iint }_{S}f\left(x(u,v),y(u,v)\right)\left|\frac{\partial (x,y)}{\partial (u,v)}\right|dudv \end{gathered}$$

line integral

piecewise-smooth curve $C$

$${\int }_{C}fds={\int }_{C}f(x(t),y(t))\sqrt{{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2}dt$$

• line integral of $f$ along $C$ with respect to $x$

  $${\int }_{C}fds={\int }_{C}f(x(t),y(t))\frac{dx}{dt}dt$$

• change orientation of $C$

  • ${\int }_{-C}fdx=-{\int }_{C}fdx$
  • ${\int }_{-C}fds={\int }_{C}fds$

work

$$W(C)={\int }_C\vec{F}\cdot d\vec{r}={\int }_C\vec{F}\cdot \vec{T}ds$$

fundamental theorem of line integral

$${\int }_C\nabla f\cdot d\vec{r}=f(\vec{r}(b))-f(\vec{r}(a))$$

conservative vector field

vector field $\vec{F}$ and $\exists f$ s.t. $\vec{F}=\nabla f$

• on open connected region, $\vec{F}=\nabla f=⟨P(x,y),Q(x,y)⟩$

  $\iff$ on simply-closed region, $\frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x}$

independence of path of line integral

$\forall C_1,C_2$ with the same ends

$${\int }_{C_1}\vec{F}\cdot d\vec{r}={\int }_{C_2}\vec{F}\cdot d\vec{r}$$

• $\iff$ closed path integral

  $${\oint }_C\vec{F}\cdot d\vec{r}=0$$

• $\iff \vec{F}$ is conservative

Green’s theorem

$C$ positively oriented, piecewise-smooth, simple closed,
$C=\partial D$ is boundary of $D$,
$P,Q$ have continuous partial derivative

$${\int }_{C}Pdx+Qdy={\iint }_D\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)dA$$

• $$A={\iint }_{D}1dA={\int }_{C}xdy=-{\int }_{C}ydx=\frac{1}{2}\int -ydx+xdy$$
• extension: multiple-connected region

nabla

$$\nabla =⟨\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z}⟩$$

Laplace operator

$${\nabla }^2=\nabla \cdot \nabla$$

curl

$$\begin{gathered} curl\vec{F}=\nabla \times \vec{F}=\left|\begin{array}{ccc}\vec{i}& \vec{j}& \vec{k}\\ \frac{\partial }{\partial x}& \frac{\partial }{\partial y}& \frac{\partial }{\partial z}\\ P& Q& R\end{array}\right| \\ =⟨\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z},\frac{\partial P}{\partial z}-\frac{\partial R}{\partial x},\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}⟩ \end{gathered}$$

• $curl(\nabla f)=\nabla \times (\nabla f)=\vec{0}$
• $\vec{F}$ conservative $\iff$ on simply-connected region, $curl\vec{F}=\vec{0}$
• irrotational $\iff curl\vec{F}=\vec{0}$

divergence

$$div\vec{F}=\nabla \cdot \vec{F}=\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}$$

• incompressible $\iff div\vec{F}=0$
• source $\iff div\vec{F}>0$
• sink $\iff div\vec{F}<0$
• $divcurl\vec{F}=\nabla \cdot (\nabla \times \vec{F})=0$
• $div(\nabla f)=\nabla \cdot (\nabla f)={\nabla }^{2}f=\frac{{\partial }^{2}f}{\partial x^2}+\frac{{\partial }^{2}f}{\partial y^2}+\frac{{\partial }^{2}f}{\partial z^2}$
  • Laplace’s equation, ${\nabla }^{2}f=0$
• for vector field, ${\nabla }^2\vec{F}={\nabla }^{2}P\vec{i}+{\nabla }^{2}Q\vec{j}+{\nabla }^{2}R\vec{k}$
• product rule, $\nabla \cdot ((\nabla f)g)=(\nabla \cdot \nabla f)g+\nabla f\cdot \nabla g$

Green’s theorem in vector form

for 2-dimension

$$\begin{gathered} {\oint }_C\vec{F}\cdot d\vec{r}={\iint }_D(curl\vec{F})\cdot d\vec{A} \\ {\oint }_C\vec{F}\cdot \vec{n}d\vec{r}={\iint }_{D}div\vec{F}d\vec{A} \end{gathered}$$

• outward unit normal vector

  $$\vec{n}=⟨\frac{y^{\prime }(t)}{|{\vec{r}}^{\prime }(t)|},-\frac{x^{\prime }(t)}{|{\vec{r}}^{\prime }(t)|}⟩$$

parametric surface

$$\vec{r}(u,v)=⟨x(u,v),y(u,v),z(u,v)⟩$$

• grid curve

• normal vector for tangent plane at $(u_0,v_0)$, ${\vec{r}}_u(u_0,v_0)\times {\vec{r}}_v(u_0,v_0)$

• smooth, ${\vec{r}}_u\times {\vec{r}}_v\ne \vec{0}$

• surface integral

  $${\iint }_{S}fdS={\iint }_{D}f\left|{\vec{r}}_u\times {\vec{r}}_v\right|dA$$

  • area

    $$A(S)={\iint }_{S}1dS={\iint }_D\left|{\vec{r}}_u\times {\vec{r}}_v\right|dA$$

  • for sphere, $dS=a^2\sin \varphi d\varphi d\theta$

oriented surface

• normal vector field

  $$\vec{n}=\frac{{\vec{r}}_u\times {\vec{r}}_v}{\left|{\vec{r}}_u\times {\vec{r}}_v\right|}$$

• $d\vec{S}=\vec{n}dS$

flux

$$\begin{gathered} {\iint }_S\vec{F}\cdot d\vec{S}={\iint }_S\vec{F}\cdot \vec{n}dS \\ ={\iint }_S\vec{F}\cdot \frac{{\vec{r}}_u\times {\vec{r}}_v}{\left|{\vec{r}}_u\times {\vec{r}}_v\right|}dS \\ ={\iint }_S\vec{F}\cdot \left({\vec{r}}_u\times {\vec{r}}_v\right)dA \end{gathered}$$

Stoke’s theorem

$$\begin{gathered} {\int }_C\vec{F}\cdot d\vec{r}={\iint }_{S}curl\vec{F}\cdot d\vec{S} \\ ={\iint }_{S}curl\vec{F}\cdot \left({\vec{r}}_u\times {\vec{r}}_v\right)dA \end{gathered}$$

divergence theorem (Gauss’s theorem)

$${\iint }_S\vec{F}\cdot \vec{n}dS={\iiint }_{E}div\vec{F}dV$$