Equations for Vector Calculus Exam 1 (Ch 1.1-3.2)

Parametric Equation

x=at+x_0

y=bt+y_0

z=ct+z_0

Resolve the Parametric Parameter

y=\left(\frac{x-x0}{a}_{}\cdot b\right)+y_0

Slope of a Parametric Line

\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{y^{\prime}\left(t\right)}{x^{\prime}\left(t\right)}

Second Derivative of a Parametric Line

\frac{d^2y}{dx^2}=\frac{\frac{d}{dt}\cdot\frac{dy}{dx}}{\frac{dx}{dt}}

Tangent Line of a Parametric Curve at (x₀, y₀)

y-y_0=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}\left(x-x_0\right)

Area under a Parametric Curve

A=\int_{a}^{b}\!y\left(t\right)\cdot x^{\prime}\left(t\right)\,dt

Arclength of a Parametric Curve

s=\int_{a}^{b}\!\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\,dt

Surface Generated by a Parametric Curve Rotated around the x-axis

SA=2\pi\int_{a}^{b}\!y\left(t\right)\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\,dt  

Rectangular to Polar Conversion

x=r\cos\left(\theta\right)

y=r\sin\left(\theta\right)

Polar to Rectangular Conversion

r=\sqrt{x^2+y^2}

\theta=\tan^{-1}\left(\frac{y}{x}\right)

Area of a Polar Curve

A=\frac12\int_{\alpha}^{\beta}\!r^2\,d\theta=\frac12\int_{\alpha}^{\beta}\!\left(f\left(\theta\right)\right)^2d\theta

Arclength of a Polar Curve

s=\int_{\alpha}^{\beta}\!\sqrt{\left(r\right)^2+\left(\frac{dr}{d\theta}\right)^2}\,d\theta=\int_{\alpha}^{\beta}\!\sqrt{\left(f\left(\theta\right)\right)^2+\left(f^{\prime}\left(\theta\right)\right)^2}\,

Distance Equation

d=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2+\left(z_2-z_1\right)^2}

Equation of a Sphere

\left(x-a\right)^2+\left(y-b\right)^2+\left(z-c\right)^2=r^2

Dot Product

X\cdot Y=\left(\left(x_1\cdot x_2\right)+\left(y_1\cdot y_2\right)+\left(z_1\cdot z_2\right)\right)

Cross Product

X\times Y=\left(x_2y_3-y_2x_3\right)i+\left(x_3y_1-x_1y_3\right)j+\left(x_1y_2-x_2y_1\right)k

Orthogonal Projection

Proj_{U}V=\frac{U\cdot V}{\left\Vert U\right\Vert^2}\cdot U

Area of a Parallelogram made by Vectors: PQ & PR

A=PQ\times PR

Area of a Triangle made by Vectors: PQ & PR

A=\frac12\left(PQ\times PR\right)

Symmetric Equation of Parametric Lines

\frac{x-x_0}{a}=\frac{y-y_0}{b}=\frac{z-z_0}{c}

Triple Scalar Product (Volume of a Parallelepiped)

U\cdot\left(V\times W\right)=\left(u_1v_2w_3+u_2v_3w_1+u_3v_1w_2\right)-\left(u_3v_2w_1+u_2v_1w_3+u_1v_3w_2\right)

Distance from a Point to a Line

d=\frac{PM\times V}{\left\Vert V\right\Vert}

Point M is given and point P is on the given vector V

Equation of a Plane

a\left(x-x_0\right)+b\left(y-y_0\right)+c\left(z-z_0\right)=0=ax+by+cz+d

Direction of the Normal Vector

N=\left(a,b,c\right)

Distance between a Point and Plane

d=\frac{\left\vert PQ\cdot N\right\vert}{\left\Vert N\right\Vert}

(Point Q is given and P is a point on the given normal vector N)

Angle Between 2 Planes

\cos\left(\theta\right)=\frac{\left\vert N_1\cdot N_2\right\vert}{\left\Vert N_1\right\Vert\left\Vert N_2\right\Vert}

Intersection of Two Planes

Put both planes into a matrix, solve the matrix via elimination, find parametric relationship relating x,y, and z. Put in a Parametric Equation

Cylindrical to Rectangular Conversion

x=r\cos\left(\theta\right) 

y=r\sin\left(\theta\right) 

z=z

Rectangular to Cylindrical Conversion

r=\sqrt{x^2+y^2}

\theta=\tan^{-1}\left(\frac{y}{x}\right)

z=z

Spherical to Rectangular Conversion

x=\rho\sin\left(\varphi\right)\cos\left(\theta\right)

y=\rho\sin\left(\varphi\right)\sin\left(\theta\right) 

z=\rho\cos\left(\varphi\right)

Spherical to Polar Conversion

r=\rho\sin\left(\varphi\right)

\theta=\theta

z=\rho\cos\left(\varphi\right)

Rectangular to Spherical Conversion

\rho=\sqrt{x^2+y^2+z^2}

\theta=\tan^{-1}\left(\frac{y}{x}\right) 

\varphi=\cos^{-1}\left(\frac{z}{\sqrt{x^2+y^2+z^2}}\right)

Polar to Spherical Conversion

\rho=\sqrt{r^2+z^2}

\theta=\theta 

\varphi=\cos^{-1}\left(\frac{z}{\sqrt{r^2+z^2}}\right)

Representation of Spherical Coordinates

rho=constant is a sphere

theta=constant is a plane

 phi=constant is a cone

Latitude and Longitude

phi=90 degrees-latitude (west is negative & vice versa)

Vector Valued Function

R\left(t\right)=f\left(t\right)i+g\left(t\right)j+h\left(t\right)k=\left(f\left(t\right),g\left(t\right),h\left(t\right)\right)

Limit of a Vector Valued Function

\lim_{t\rightarrow a}R\left(t\right)=\left(\lim_{t\rightarrow a}f\left(t\right),\lim_{t\rightarrow a}g\left(t\right),\lim_{t\rightarrow a}h\left(t\right)\right)

Derivative of a Vector Valued Function

\frac{d}{dt}R\left(t\right)=\left(\frac{d}{dt}f\left(t\right),\frac{d}{dt}g\left(t\right),\frac{d}{dt}h\left(t\right)\right)

Integral of a Vector Valued Function

\int_{a}^{b}\!R\left(t\right)\,dt=\left(\int_{a}^{b}\!f\left(t\right)\,dt,\int_{a}^{b}\!g\left(t\right)\,dt,\int_{a}^{b}\!h\left(t\right)\,dt\right)

Unit Tangent Vector of a Vector Valued Function

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