Calculus

Directional Derivative

Multivariate

Line Integral

$\int_{C}f(x,y)ds=\int_Cf(x(t), y(t))|r'(t)|dt$

Partial Differentiation

Multiple Integration

Vector fields

Surface Integral

Green's Theorem

Stroke's Theorem

Divergence Theorem

Applications

Center of Mass

Mass/Charge Density

if $z=g(x,y)$

Chain Rule

Case 1
$f(x,y), x=g(t), y=h(t)$

Case 2
$f(x,y), x=g(u,v), y=h(u,v)$

$\frac{df}{dt}=\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}$

$\frac{df}{du}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial u}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial u}$

$\frac{df}{dv}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial v}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial v}$

Implicit Function Theorem
$f(x,y)=0$

$\frac{dy}{dx}=-\frac{\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial y}}=-\frac{F_x}{F_y}$

$\frac{\partial z}{\partial x}=-\frac{\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial z}}$

$\frac{\partial z}{\partial y}=-\frac{\frac{\partial F}{\partial y}}{\frac{\partial F}{\partial z}}$

Gradient

Directional Derivative

$\nabla f=\langle f_x, f_y, f_z\rangle$

$D_uf=\nabla f*u$

Maximum and Minimum

Second Derivative Test

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

$D>0$ and $f_{xx}>0$ -- local minimum
$D>0$ and $f_{xx}<0$ -- local maximum
$D<0$ -- saddle point

Lagrange Multiplier

$\nabla f = \lambda \nabla g$

Double Integrals

Triple Integrals

Type 1

Type 2

Polar Coordinate

$\int\int_D f(x,y) dydx$

$\int\int_D f(x,y) dxdy$

$\int\int_D f(rcos\theta, rsin\theta) rdrd\theta$

$(\tilde x, \tilde y)$

$\tilde x = \frac{M_y}{m}$, $\tilde y = \frac{M_x}{m}$

$M_x=\int\int_Dx\rho (x,y)dA$
$M_y=\int\int_Dy\rho (x,y)dA$

$m=\int\int_D\rho (x,y)dA$

lamina

Surface Area

$z=f(x,y)$

$\int\int_D \sqrt {1+z_x^2+z_y^2}dA$

Regular

Cylindrical

Spherical

$\int\int\int_E f(x,y,z) dzdydx$

$\int\int\int_E f(rcos\theta, rsin\theta, z) rdzdrd\theta$

$\int\int\int_E f(\rho sin\phi cos\theta, \rho sin\phi sin\theta, \rho cos\phi) \rho^2sin(\phi)d\rho d\phi d\theta)$

$ \int_{C}F(x,y)\cdot dr=\int_C F(x(t), y(t))\cdot r'(t)dt$

$ \int\int_{S}f(x,y,z)dS=\int\int_{D}f(x,y,z)\sqrt{1+g_x^2+g_y^2}dA$

(upward orientation or normal changes sign)
$ \int\int_{S}F(x,y,z)\cdot dS=\int\int_{D}F(x,y,z) \cdot \langle -g_x, -g_y, 1\rangle dA$
(simple surface)
$ \int\int_{S}F(x,y,z)\cdot dS=\int\int_{D}F(x,y,z) \cdot ndA$

else S is $r(u,v)$

$ \int\int_{S}f(x,y,z)dS=\int\int_{D}f(r(u,v))|r_u\times r_v|dA$

$ \int\int_{S}F(x,y,z)\cdot dS=\int\int_{D}F(r(u,v)) \cdot (r_u\times r_v)dA$

Fundamental Theorem of Line Integral

if $F=\nabla f$

$\int_c F\cdot dr=f(r(b)) - f(r(a))$

Closed piece-wise line C and
D is area inside (2D)

$\oint_c F\cdot dr=\int\int_D(Q_x-P_y)dA$

Deriving Scalar Potential Function $f$

Closed piece-wise line C (3D)

$\oint_c F\cdot dr=\int\int_S curlF dS$

Closed surface S

$\int\oint_cF \cdot dS=\int\int\int_EdivFdV$