multivariable calc

Vector-valued function

function mapping real inputs to vector outputs

Level curve

curve where a function f(x, y) = c is constant

Gradient vector

vector of all partial derivatives of a scalar function

Directional derivative

rate of change of a function in a specific direction

Vector-valued function

function mapping real inputs to vector outputs

Level curve

curve where a function f(x, y) = c is constant

Gradient vector

vector of all partial derivatives of a scalar function

Directional derivative

rate of change of a function in a specific direction

Divergence

scalar measure of a vector field's outward flow \nabla \cdot F

Curl

vector measure of rotation \nabla \times F in a field

Chain rule

derivative of a composite function using partial derivatives

Jacobian matrix

matrix of first-order partial derivatives of a vector function

Second partial derivative test

method to classify critical points using D = f{xx} f{yy} - f_{xy}^2

Local minimum condition

D > 0, f_{xx} > 0

Local maximum condition

D > 0, f_{xx} < 0

Saddle point condition

D < 0

Equation of a plane

a(x - x0) + b(y - y0) + c(z - z_0) = 0

Tangent plane

linear approximation to surface at a point using partial derivatives

Total differential

approximation: dz = fx dx + fy dy

Scalar field

function assigning a scalar to each point in space

Vector field

function assigning a vector to each point in space

Line integral (scalar)

\int_C f(x, y) ds, integration over arc length

Line integral (vector)

\int_C F \cdot dr, work done by a field

Surface integral (scalar)

\iint_S f(x, y, z) dS, sum over a surface

Surface integral (vector)

\iint_S F \cdot dS, flux through a surface

Green

’s Theorem

relates line integral around a curve to a double integral over the region

Stokes’ Theorem

relates line integral to surface integral of the curl

Divergence Theorem

relates flux over closed surface to triple integral of divergence

Line equation in space

r(t) = r_0 + tv

Dot product

scalar product: a \cdot b = |a||b|\cos\theta

Cross product

vector perpendicular to both input vectors

Volume under a surface

\iint_R f(x, y) dA

Triple integral (Cartesian)

\iiint_E f(x, y, z) dV

Cylindrical coordinates

x = r \cos\theta, y = r \sin\theta, z = z

Spherical coordinates

x = \rho \sin\phi \cos\theta, y = \rho \sin\phi \sin\theta, z = \rho \cos\phi

Jacobian (polar)

r

Jacobian (spherical)

\rho^2 \sin\phi

Conservative vector field

vector field that equals gradient of a scalar function

Test for conservativeness

\nabla \times F = 0

Path independence

property of conservative fields where integral depends only on endpoints

Taylor expansion (2 variables)

polynomial approximation of multivariable function

Lagrange multipliers

method to optimize a function with constraints using \nabla f = \lambda\nabla g

Implicit function theorem

conditions under which one variable can be solved in terms of others

Change of variables

method using substitution and Jacobian determinant

Tangent vector to curve

derivative r'(t)

Normal vector to surface

cross product of two tangent vectors

Arc length formula

\int_a^b |r'(t)| dt

Unit tangent vector

normalized derivative T(t) = r'(t)/|r'(t)|

Curvature

rate of change of the unit tangent vector with respect to arc length

Torsion

measure of how a curve twists out of its plane

Triple integral (cylindrical)

\iiint_E f(r, \theta, z) r dz dr d\theta

Triple integral

\iiint_E f(\rho, \phi, \theta) \rho^2 \sin\phi d\rho d\phi d\theta