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Calculus (Multivariate (Partial Differentiation (Chain Rule (Implicit…

- - - - Case 1
        \(f(x,y), x=g(t), y=h(t)\)
        
        \(\frac{df}{dt}=\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}\)
      - Case 2
        \(f(x,y), x=g(u,v), y=h(u,v)\)
        
        \(\frac{df}{du}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial u}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial u}\)
        \(\frac{df}{dv}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial v}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial v}\)
      - Implicit Function Theorem
        \(f(x,y)=0\)
        
        \(\frac{dy}{dx}=-\frac{\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial y}}=-\frac{F_x}{F_y}\)
        
        \(\frac{\partial z}{\partial x}=-\frac{\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial z}}\)
        \(\frac{\partial z}{\partial y}=-\frac{\frac{\partial F}{\partial y}}{\frac{\partial F}{\partial z}}\)
    - - \(\nabla f=\langle f_x, f_y, f_z\rangle\)
    - - \(D_uf=\nabla f*u\)
    - - 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
        
        \(\nabla f = \lambda \nabla g\)
  - - - Type 1
        
        \(\int\int_D f(x,y) dydx\)
      - Type 2
        
        \(\int\int_D f(x,y) dxdy\)
      - Polar Coordinate
        
        \(\int\int_D f(rcos\theta, rsin\theta) rdrd\theta\)
      - Surface Area
        
        \(z=f(x,y)\)
        \(\int\int_D \sqrt {1+z_x^2+z_y^2}dA\)
    - - Regular
        
        \(\int\int\int_E f(x,y,z) dzdydx\)
      - Cylindrical
        
        \(\int\int\int_E f(rcos\theta, rsin\theta, z) rdzdrd\theta\)
      - Spherical
        
        \(\int\int\int_E f(\rho sin\phi cos\theta, \rho sin\phi sin\theta, \rho cos\phi) \rho^2sin(\phi)d\rho d\phi d\theta)\)
  - - - \( \int\int_{S}f(x,y,z)dS=\int\int_{D}f(x,y,z)\sqrt{1+g_x^2+g_y^2}dA\)
        (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)
        \( \int\int_{S}F(x,y,z)\cdot dS=\int\int_{D}F(x,y,z) \cdot ndA\)
    - - \( \int\int_{S}f(x,y,z)dS=\int\int_{D}f(r(u,v))|r_u\times r_v|dA\)
        \( \int\int_{S}F(x,y,z)\cdot dS=\int\int_{D}F(r(u,v)) \cdot (r_u\times r_v)dA\)
  - - - \(\oint_c F\cdot dr=\int\int_D(Q_x-P_y)dA\)
  - - - \(\oint_c F\cdot dr=\int\int_S curlF dS\)
  - - - \(\int\oint_cF \cdot dS=\int\int\int_EdivFdV\)
  - - - \((\tilde x, \tilde y)\)
        
        \(\tilde x = \frac{M_y}{m}\), \(\tilde y = \frac{M_x}{m}\)
        
        \(M_x=\int\int_Dx\rho (x,y)dA\)
        \(M_y=\int\int_Dy\rho (x,y)dA\)
    - - \(m=\int\int_D\rho (x,y)dA\)
  - - - \(\int_c F\cdot dr=f(r(b)) - f(r(a))\)