AMS 261 Equations

component form

v = <q₁-p₁, q₂-p₂> = <v₁,v₂>

magnitude

||v|| = sqrt(v₁²+v₂²+v₃²)

vector addition

u + v = <u₁+v₁,u₂+v₂>

scalar multiplication

cu = <cu₁,cu₂>

unit vector

• a vector with a magnitude of 1

• u = v / ||v||

triangle inequality

||u+v|| </= ||u|| + ||v||

distance

• the distance between points (x₁,y₁,z₁) and (x₂,y₂,z₂)

• d = sqrt[(x₂-x₁)²+(y₂-y₁)²+(z₂-z₁)²]

midpoint formula

• average of two points

• M = [(x₁+x₂ / 2), y₁+y₂ / 2), z₁+z₂ / 2)]

equation of a sphere

(x-x₁)² + (y-y₁)² + (z-z₁)²= r²

dot product

u ⋅ v = u₁v₁+u₂v₂+u₃v₃

angle between two vectors

cosθ = u ⋅ v / ||u|| ||v||

vector projection

proj_vu = (u ⋅ v / ||v||²) * v

Work

• the work W done by a constant force F moving an object along the vector PQ

• W = F⋅ PQ

Cross product

u x v = |matrix|=(u₂v₃ - u₃v₂) i - (u₁v₃ - u₃v₁) j + (u₁v₂ - u₂v₁) k

*the resulting vector u x v is orthogonal to both u and v

geometric idea

• the magnitude of the cross product is the area of the parallelogram

• ||u x v|| = ||u|| ||v|| sin\theta

triple scalar product

V = | u ⋅ (v x w) |

parametric equations

x = x₀ + at, y = y₀ + bt, z = z₀ + ct

symmetric equations

(x - x₀) / a = (y - y₀) / b = (z - z₀) / c

standard (point-normal) form

a(x - x₀) + b(y - y₀) + c(z - z₀) = 0

general form

ax + by + cz + d = 0

distance between a point and a plane

• point P (x₀,y₀,z₀) to the plane ax + by + cz + d = 0

• D = |ax₀+by₀+cz₀+d| / \sqrt{a^2+b^2+c^2}

distance between a point and a line in space

D = ||PQ x v|| / ||v||

ellipsoid

\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1

hyperboloid of one sheet

\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1

hyperboloid of two sheets

-\frac{x^2}{a^2}-\frac{y^2}{b^2}+\frac{z^2}{c^2}=1

elliptic cone

\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{z^2}{c^2}

elliptic paraboloid

z=\frac{x^2}{a^2}+\frac{y^2}{b^2}

hyperbolic paraboloid

z=\frac{y^2}{b^2}-\frac{x^2}{a^2}

coordinate conversion

cylindrical → rectangular: x = rcos\theta, y=rsin\theta, z=z

rectangular → cylinderical: r=\sqrt{x^2+y^2,}\tan\theta=\frac{y}{x},z=z

spherical → rectangular: x=\rho\sin\phi\cos\theta,y=\rho\sin\phi\sin\theta,z=\rho\cos\phi

rectangular → spherical: \rho=\sqrt{x^2+y^2+z^2},\tan\theta=\frac{y}{x},\phi=\arccos\left(\frac{z}{\sqrt{r^2+z^2}}\right)

cylindrical → spherical:

\rho=\sqrt{r^2+z^2},\theta=\theta,\phi=\arccos\left(\frac{z}{\sqrt{r^2+z^2}}\right)

spherical → cylindrical:r=\rho\sin\phi,\theta=\theta,z=\rho\cos\phi

projectile motion

r(t) = (-1/2gt²) j + tv₀ + r₀

unit tangent vector

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

principal unit normal vector

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

T\left(t\right)\cdot N\left(t\right)

partial derivatives

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

f_x(x,y) = \frac{\delta z}{\delta x}

f_y(x,y) = \frac{\delta z}{\delta y}

limit definition partial derivatives

f_x(x,y) =

\lim_{\Delta x\to0}\frac{f\left(x+\Delta x,y\right)-f\left(x,y\right)}{\Delta x}

f_y(x,y) =

\lim_{\Delta y\to0}\frac{f\left(x,y+\Delta y\right)-f\left(x,y\right)}{\Delta y}

mixed partials

f_xy(x,y)=f_yx(x,y)

total differential

dz=fx\left(x,y\right)dx+fy\left(x,y\right)dy=\frac{\delta z}{\delta x}dx+\frac{\delta z}{\delta y}dy

approximation of differentials

\Delta z=fx\left(x0,y0\right)\Delta x+fy\left(x0,y0\right)\Delta y

chain rule

\frac{\delta z}{\delta s}=\frac{\delta z}{\delta x}\frac{\delta x}{\delta s}+\frac{\delta z}{\delta y}\frac{\delta y}{\delta s}

\frac{\delta z}{\delta t}=\frac{\delta z}{\delta x}\frac{\delta x}{\delta t}+\frac{\delta z}{\delta y}\frac{\delta y}{\delta t}

implicit partial derivatives

\frac{dy}{dx}=-\frac{Fx}{Fy}

gradient

∇f(x,y)=<f_x(x,y),f_y(x,y)>=f_x i + f_y j

directional derivative

• D_uf = ∇f⋅u = f_xa + f_yb + f_zc

• D_uf = ||∇f||cosθ

tangent plane

F_x(P)(x-x₀)+F_y(P)(y-y₀)+F_z(P)(z-z₀)=0

second partials test

D = f_xx(a,b)f_yy(a,b)-[f_xy(a,b)]²

lagrange multipliers

\nabla f\left(x,y,z\right)=\lambda\nabla g\left(x,y,z\right) and g(x,y,z) = k

• f_x= \lambda g_x

• f_y= \lambda g_y

• f_z= \lambda g_z

• g(x,y,z)=k

iterated integrals

\int\int_{R}^{}\!f\left(x,y\right)\,dA

area a region R

A=\int\int_{R}^{}\!1\,dA

volume of a solid

V=\int\int_{R}^{}\!f\left(x,y\right)\,dA

integrals over dy dx

\int\int_{R}^{}\!f\left(x,y\right)dA=\int_{a}^{b}\!\int_{g1\left(x\right)}^{g2\left(x\right)}\!f\left(x,y\right)\,dydx

integrals over dx dy

\int\int_{R}^{}\!f\left(x,y\right)dA=\int_{c}^{d}\!\int_{h1\left(x\right)}^{h2\left(x\right)}\!f\left(x,y\right)\,dxdy

average value of a function

f_avg=

\frac{1}{Area\left(R\right)}\int\int_{R}^{}\!f\left(x,y\right)\,dA

change of variables to polar coordinates

x=r\cos\theta

y=r\sin\theta

x^2+y^2=r^2

dA=rdrd\theta

double integral polar form

\int\int_{R}^{}\!f\left(x,y\right)\,dA=\int\int_{R}^{}\!f\left(r\cos\theta,r\sin\theta\right)r\,drd\theta

vector curl

curlF\left(x,y,z\right)=\nabla x F(x,y,z)

vector divergence

∇ ⋅ F(x,y,z) =

\frac{\delta M}{\delta x}+\frac{\delta N}{\delta y}+\frac{\delta P}{\delta z}