Multivariable Calculus Formulas Flashcards
Equation of a tangent plane to z = f(a,b)
z = f(a,b) + df/dx(a,b).(x - a) + df/dy(a,b).(y - b)
Classification of critical points in the (x,y) plane
- |H| > 0, fxx < 0 -> local maximum
- |H| > 0, fxx > 0 -> local minimum
- |H| < 0 -> saddlepoint
- |H| = 0 -> unknown
Change of variables:
dxdy -> dudv
dxdy = |d(x,y)/d(u,v)| dudv
Change of variables:
dxdy -> drd(theta)
dxdy = r * drd(theta)
Change of variables:
dxdydz -> drd(theta)d(phi)
dxdydz = r^2*sin(theta) * drd(theta)d(phi)
Change of variables:
(x, y) -> (r, theta)
x = r * cos(theta)y = r * sin(theta)
Change of variables:
(x, y, z) -> (r, theta, phi)
x = r * sin(theta) * cos(phi)y = r * sin(theta) * sin(phi)z = r * cos(theta)
Canonical form of a circle
x^2 + y^2 = r^2
Canonical form of an elipse
x^2/a^2 + y^2/b^2 = r^2
Canonical form of a hyperbolae
x^2/a^2 - y^2/b^2 = r^2
How do you determine if a function from (x, y) -> (u, v) is invertible in (u, v)
Invertibility theorem:d(x, y) / d(u, v) != 0
Line integral equation equal to the area of an enclosed (simple-connected) domain
Direction and magnitude of steepest ascent
Direction of delta.f = (df/dx.i + df/dy.j)
Magnitude of delta.f
Taylor expansion of f(x, y) about (a, b)
Directional derivative
D(u) = (df/dx*i + df/dy*j).(ai + bj)/|ai + bj|
Cos double angle formula
cos(2*theta) = cos^2(theta) - sin^2(theta)
Sin double angle formula
sin(2*theta) = 2*cos(theta)*sin(theta)
Direction of steepest ascent
v = df/dx*i + df/dy*j
Directions of constant height
Two directions orthogonal to v, the direction of steepest ascent (+,-)
Path of steepest ascent
dy/dx = df/dy / df/dx
J(u,v)
d(x,y) / d(u,v)
Green’s theorem
Lagrange multiplier equations
d(f,g)/d(x,y) = 0
g(x,y) = 0 (constraint)
