Equations for Vector Calculus Exam 1 (Ch 3.3-?.?)

Note that any variable that is capitalized is a vector

Arclength for R(t)=(f(t), g(t), h(t))

s\left(t\right)=\int_{a}^{b}\!\sqrt{\left(f^{\prime}\left(t\right)\right)^2+\left(g^{\prime}\left(t\right)\right)^2+\left(h^{\prime}\left(t\right)\right)^2}\,dt

Arclength Parameterization for R(t)=(f(t), g(t), h(t))

s\left(t\right)=\int_{a}^{t}\!\sqrt{\left(f^{\prime}\left(u\right)\right)^2+\left(g^{\prime}\left(u\right)\right)^2+\left(h^{\prime}\left(u\right)\right)^2}\,du

3 Dimensional Curvature

K=\frac{\left\Vert T^{\prime}\left(t\right)\right\Vert}{\left\Vert R^{\prime}\left(t\right)\right\Vert}=\frac{\left\Vert R^{\prime\prime}\left(t\right)\times R^{\prime}\left(t\right)\right\Vert}{\left\Vert R^{\prime}\left(t\right)\right\Vert^3}

2 Dimensional Curve

C=\frac{\left\vert y^{\prime\prime}\left(x\right)\right\vert}{\left(1+\left\vert y^{\prime}\left(x\right)\right\vert^2\right)^{\frac32}}

Principal Normal Vector

N\left(t\right)=\frac{T^{\prime}\left(t\right)}{\left\vert T^{\prime}\left(t\right)\right\vert}

Unit Binormal Vector

B\left(t\right)=T\left(t\right)\times N\left(t\right)

Position of R(t)

R\left(t\right)=\int V\left(t\right)dt=\iint A\left(t\right)dtdt

Velocity of R(t)

V\left(t\right)=R^{\prime}\left(t\right)=\int A\left(t\right)dt

Acceleration of R(t)

A\left(t\right)=V^{\prime}\left(t\right)=R^{\prime\prime}\left(t\right)

Speed of R(t)

Speed=\left\vert V\left(t\right)\right\vert=\left\vert R^{\prime}\left(t\right)\right\vert

Level Curves

f\left(x,y\right)=c

Vertical Traces (shows yz plane)

f\left(a,y\right)=c

Horizontal Traces (shows xz plane)

f\left(x,b\right)=c

Level Surface

w\left(x,y,z\right)=c

Multivariable Limit Conditions (The limit exists if…)

All potential paths lead to the same existing point

\lim_{\left(x,b\right)\rightarrow\left(a,b\right)}f\left(x,y\right)=\lim_{\left(a,y\right)\rightarrow\left(a,b\right)}f\left(x,y\right)

Tangent Plane Equation

z=f\left(x_0,y_0\right)+f_{x}\left(x_0,y_0\right)\left(x-x_0)+f_{y}\left(x_0,y_0\right)\left(y-y_0\right)\right) 

N=\left(j\times f_{y}\left(x_0,y_0\right)k\right)\times\left(i\times f_{x}\left(x_0,y_0\right)k\right)

Linear Approximation Equation

z=f\left(x_0,y_0\right)+f_{x}\left(x_0,y_0\right)\left(x-x_0)+f_{y}\left(x_0,y_0\right)\left(y-y_0\right)\right)

Chain Rule

\frac{dz}{dt}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial t},z=f\left(x\left(t\right),y\left(t\right)\right)

Implicit Differentiation for f(x,y) and y=g(x)

\frac{dy}{dx}=-\frac{\frac{\partial f}{\partial x}}{\frac{\partial f}{\partial y}},\frac{dz}{dx}=-\frac{\frac{\partial f}{\partial x}}{\frac{\partial f}{\partial z}}

Directional Derivatives

D_{u}=\nabla f\left(x,y\right)\cdot\begin{pmatrix}\cos\left(\theta\right)\\ \sin\left(\theta\right)\end{pmatrix}=\nabla f\left(x,y\right)\cdot\frac{V}{||V||}=\left\Vert\nabla f\left(x,y\right)\right\Vert\cdot\cos\left(\theta\right)=\nabla f\left(x,y\right)\cdot U=\nabla f\left(x,y\right)\cdot\begin{pmatrix}u_{x}\\ u_{y}\\ u_{z}\end{pmatrix}

Critical Point

\nabla f(x,y)=0

Types of Extrema (Absolute Extrema is only at boundaries and critical points)

D=f_{xx}\left(x_0,y_0\right)f_{yy}\left(x_0,y_0\right)-\left(f_{xx}\left(x_0,y_0\right)\right)^2

If D>0 & fxx>0, it’s a minimum

If D>0 & fxx<0, it’s a maximum

If D<0, it’s a saddle point

If D=0 it fails

2 Dimensional Lagrange Multiples

\begin{pmatrix}\nabla f\left(x_0,y_0\right)\\ g\left(x_0,y_0\right)\end{pmatrix}=\begin{pmatrix}\lambda\nabla g\left(x_0,y_0\right)\\ 0\end{pmatrix}

3 Dimensional Lagrange Multiples

\begin{pmatrix}\nabla f\left(x_0,y_0\right)\\ g\left(x_0,y_0\right)\\ h\left(x_0,y_0\right)\end{pmatrix}=\begin{pmatrix}\lambda_1\nabla g\left(x_0,y_0\right)+\lambda_2\nabla h\left(x_0,y_0\right)\\ 0\\ 0\end{pmatrix}

Type 1 Double Integral (Vertically Simple)

\iint f\left(x,y\right)dA=\int_{a}^{b}\int_{g_1\left(x\right)}^{g_2\left(x\right)}f\left(x,y\right)dydx

Type 2 Double Integral (Horizontally Simple)

\iint f\left(x,y\right)dA=\int_{c}^{d}\int_{h_1\left(y\right)}^{h_2\left(y\right)}f\left(x,y\right)dxdy

Polar Double Integrals

\iint f\left(x,y\right)dA=\int_{\alpha}^{\beta}\int_{h_1\left(\theta\right)}^{h_2\left(\theta\right)}f\left(r,\theta\right)rdrd\theta

Cartesian Triple Integrals (Could be rewritten for any integration order)

\int_{a}^{b}\int_{v_1\left(x\right)}^{v_2\left(x\right)}\int_{u_1\left(x,y\right)}^{u_2\left(x,y\right)}f\left(x,y,z\right)dzdydx

Cylindrical Triple Integrals (Could be rewritten for any integration order)

\int_{\alpha}^{\beta}\int_{v_1\left(\theta\right)}^{v_2\left(\theta\right)}\int_{u_1\left(r,\theta\right)}^{u_2\left(r,\theta\right)}f\left(r,\theta,z\right)rdzdrd\theta

Spherical Triple Integrals (Could be rewritten for any integration order)

\int_{\psi}^{\delta}\int_{v_1\left(\varphi\right)}^{v_2\left(\varphi\right)}\int_{u_1\left(\theta,\varphi\right)}^{u_2\left(\theta,\varphi\right)}f\left(\rho,\theta,\varphi\right)\rho^2\sin\left(\varphi\right)d\rho d\theta d\varphi

2 Dimensional Total Mass

m=\iint_{R}\rho\left(x,y\right)dA

3 Dimensional Total Mass

m=\iiint_{Q}\rho\left(x,y\right)dV

Moment About the X-Axis

M_{x}=\iint_{R}\rho\left(x,y\right)ydA

Moment About the Y-Axis

M_{y}=\iint_{R}\rho\left(x,y\right)xdA

2 Dimensional Center of Mass

\overline{x}=\frac{M_{y}}{m},\overline{y}=\frac{Mx}{m}_{}

2 Dimensional X-Axis Moment of Inertia

I_{x}=\iint_{R}\rho\left(x,y\right)y^2dA

2 Dimensional Y-Axis Moment of Inertia

I_{y}=\iint_{R}\rho\left(x,y\right)x^2dA

2 Dimensional Polar Moment of Inertia

I=I_{x}+I_{y}

Moment About the XY-Plane

M_{xy}=\iiint_{Q}\rho\left(x,y\right)zdV

Moment About the YZ-Plane

M_{yz}=\iiint_{Q}\rho\left(x,y\right)xdV

Moment About the XZ-Plane

M_{xz}=\iiint_{Q}\rho\left(x,y\right)ydV

3 Dimensional Center of Mass

\overline{x}=\frac{M_{yz}}{m},\overline{y}=\frac{M_{xz}}{m},\overline{z}=\frac{M_{xy}}{m}_{}

3 Dimensional X-Axis Moment of Inertia 

I_{x}=\iiint_{Q}\left(y^2+z^2\right)\rho\left(x,y\right)dV

3 Dimensional Y-Axis Moment of Inertia

I_{y}=\iiint_{Q}\left(x^2+z^2\right)\rho\left(x,y\right)dV

3 Dimensional Z-Axis Moment of Inertia

I_{z}=\iiint_{Q}\left(x^2+y^2\right)\rho\left(x,y\right)dV

Change in Variable

\left(x,y\right)=T\left(u,v\right)=\left(g\left(u,v\right),h\left(u,v\right)\right)

Jacobian

J=\frac{\partial\left(x,y\right)}{\partial\left(u,v\right)}=\begin{pmatrix}\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v}\\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v}\end{pmatrix}

Change in Variable in Multiple Integration

\iint_{R}f\left(x,y\right)dA=\iint_{S}f\left(u,v\right)\left\vert J\right\vert dudv