calculus (advanced)

vector-valued functions

differentiating a vector-valued function by derivating of each of the components

(vector-valued function derivatives) sum and difference property

(vector-valued function derivatives) scalar product property

(vector-valued function derivatives) dot product property

(vector-valued function derivatives) cross product property

(vector-valued function derivatives) chain product property

r(t) ⋅ r(t) = constant ⇒ r(t) ⋅ r'(t) = 0

arc length function and it's derivative

curvature formulas

radius of curvature

differentiation of functions of several variables

Clairaut's theorem

equation of a tangent plane

understanding differentiability for functions of two variables

If z = f(x,y), then f is differentiable at (a, b) if Δz can be expressed in the form [Δz = f_x(a, b) Δx + f_y(a, b) Δy + ε₁ Δx + ε₂ Δy]

differentiability implies continuity (incomplete)

continuity of first partials implies differentiability (incomplete)

chain rule for one independent variable

chain rule for two independent variable (incomplete)

generalized chain rule (incomplete)

implicit differentiation with two variables

implicit differentiation with three variables

D_u f(x,y) = f_x(x,y) cos(θ) + f_y(x,y) sin(θ)

gradient is normal to the level curve

gradient vector is perpendicular to the tangent vector to any curve on a surface

second derivative test (incomplete)

multiple integrals

definition of double integrals

properties of double integrals (incomplete)

iterated integral (incomplete)

Fubini's theorem (incomplete)

Fubini's Theorem (Strong Form) (incomplete)

Fubini's theorem for improper integrals (incomplete)

integrating over polar rectangles

integrating over general polar regions (incomplete)

area formula for polar regions (incomplete)