Calc 3 Final
Chain rule strategy
Build a chart of functions, determine which partials to take
D =
D = fxx(a,b)fyy(a,b) - [fxy(a,b)]^2 D>0 and fxx>0, relative min D>0 and fxx<0, relative max D=0, inconclusive D<0, saddle point
Limits of multivariable functions
Replace y or x with each other or t, find a way to make the limit equal two different values
Critical points of f(x,y)
Wherever fx (∂f/∂x) and fy (∂f/∂y) = 0
Equation of a plane given a point P (x0, y0, z0) and a normal vector n <a, b, c>
a(x-x0) + b(y-y0) + c(z-z0) = 0 or a(x) + b(y) + c(z) = d
div(curlF) = ?
div(curlF) = 0
divF where F = <P(x,y,z), Q(x, y, z), R(x, y, z)>
divF = ∇ ⋅ F = ∂P/∂x + ∂Q/∂y + ∂R/∂z (scalar) If divF = 0, source free
Planes are parallel if Planes are orthogonal if
n1 and n2 are parallel n1 ⋅ n2 = 0
Surface integral (vector field) where F = <f, g, h>
s∫∫ F (f,g,h) dS = R∫∫(-(f* hx) -(g* hy) +h) dA
Surface integral (vector field) where F (r(t))
s∫∫ F (r(u,v)) dS = R∫∫ F ⋅ (∂r/∂u x ∂r/∂v) dA
Equation of a plane tangent to a surface z = f(x,y) and linear approximation
z = f(x0,y0) + fx(x0,y0)(x-x0) + fy(x0,y0)(y-y0)
dV in Spherical Coordinates
ρ^2 sinφ dρ dφ dθ ρ: radial distance from center φ: Angle from horizontal plane
Vector field F = <P, Q, R> is conservative if
∂P/∂y = ∂Q/∂x } R2 ∂P/∂z = ∂R/∂x ∂Q/∂z = ∂R/∂y
∇
∇ = (∂/∂x)î (∂/∂y)ĵ (∂/∂z)k̂
∇f
∇f = (fx, fy, fz) Points in direction of steepest ascent |∇f| = Rate of increase in direction of steepest ascent at a certain point
To find potential function f (x, y, z) of vector field F = <P, Q, R>:
1. ∫P(x, y, z) dx, OR ∫Q(x, y, z) dy, OR ∫R(x, y, z) dz 2. ∂/∂y = Q (x, y, z) to find Cy 3. Integrate Cy with respect to y 4. ∂/∂z = R (x, y, z) to find d'(z) 5. Integrate d'(z) with respect to z
Area using Green's theorem
A = c∮x dy = -c∮y dx = 1/2* c∮xdy-ydx
Arc Length of a vector r(t) = <f(t), g(t), h(t)>
a∫b |r'(t)|
curl F where F = <P(x,y,z), Q(x, y, z), R(x, y, z)>
curlF = ∇ x F curlF (R2) = ∂Q/∂x - ∂P/∂y (vector) If curlF = 0, irrotational
Line integral formula (vector fields)
c∫ F(x,y) dr = a∫b F(r(t)) ⋅ r'(t) dt where r(t) = (1-t)(x1,y1) + t(x2,y2) a=0, b=1 or r(t) = length of curve (circle, etc)
Line integral formula (vector valued function)
c∫ f(x,y) ds = a∫b f(h(t),g(t)) * |r'(t)| dt where r(t) = (1-t)(x1,y1) + t(x2,y2) a=0, b=1 or r(t) = length of curve (circle, etc) Piecewise curves: c∫ f(x,y) ds = c1∫ +c2∫ +c3∫ ...
Green's Theorem
c∮F dr = D∫∫curlF dA c ∮F dr = D∫∫curlF dA
Chain rule Given z = f(x, y) where x = h(t) and y = g(t), find dz/dt
dz/dt = ∂f/∂x* dx/dt + ∂f/∂y* dy/dt
dA in Polar Coordinates
r dr dθ
Parametric Description of a cylinder r(u,v)
r(u,v) = <a* cosu, a* sinu, v) a:radius 0≤u≤2pi 0≤v≤h
Parametric Description of a sphere r(u,v)
r(u,v) = <a* sinu* cosv, a* sinu* sinu, a* cosu> a: radius 0≤u≤1/2pi 0≤v≤2pi
Surface integral (vector valued function) where z = g(x,y)
s∫∫ f (x,y,z) dS = D∫∫f (x,y,g(x,y))* ((∂g/∂x)^2 + (∂g/∂y)^2+1))^1/2 dA
Surface integral (vector valued function) where f(r(u,v))
s∫∫ f (x,y,z) dS = D∫∫f(r(u,v) | ru x rv | dA
Cross Product
u x v = (u2v3 - u3v2)î + (u3v1 - u1v3)ĵ + (u1v2 - u2v1)k̂ Right hand rule defines direction (u x v) ⋅ v = (u x v) ⋅ u = 0
Dot Product
u ⋅ v = u1v1 + u2v2 + u3v3 If u ⋅ v = 0, u and v are perpendicular.
