The multivariable chain rule finds the derivative of a function whose inputs depend on other variables. If z = f(x, y) with x = x(t) and y = y(t), then dz/dt = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt). Each path from z down to t in a tree diagram gives one product, and you add the products.
A tangent plane to the surface z = f(x, y) at (a, b) is the plane that touches the surface there and has the same slopes in the x and y directions. Its equation is z = f(a, b) + f_x(a, b)(x − a) + f_y(a, b)(y − b). Near the point, the plane approximates the function.
A partial derivative measures how a function of several variables changes when one input changes and the others stay fixed. To find ∂f/∂x, treat y as a constant and differentiate with respect to x. To find ∂f/∂y, treat x as a constant. Each partial derivative is the slope of the surface in one direction.
The limit of a multivariable function f(x, y) as (x, y) → (a, b) is L when f(x, y) gets close to L no matter which path (x, y) takes toward (a, b). For a continuous function you can substitute directly. To show a limit does not exist, find two paths that give different values.
A function of several variables, such as z = f(x, y), assigns one output to each input point (x, y). Its domain is where the formula is defined, its graph is a surface in 3D, and its level curves f(x, y) = k, drawn together as a contour map, show that surface from above like a hiking map.
Motion in space describes a moving object with vectors: position r(t), velocity v(t) = r′(t), speed ‖v(t)‖ and acceleration a(t) = r″(t). Acceleration splits into a tangential part (v · a)/‖v‖ that changes the speed and a normal part ‖v × a‖/‖v‖ that changes the direction.
Curvature, written κ (kappa), measures how sharply a curve bends. It is the rate at which the unit tangent vector turns per unit of distance: κ = ‖T′(t)‖/‖r′(t)‖ = ‖r′ × r″‖/‖r′‖³. A straight line has curvature 0, and a circle of radius R has curvature 1/R.
The unit tangent vector T(t) = r′(t)/‖r′(t)‖ points in the direction a curve is moving, and the unit normal vector N(t) = T′(t)/‖T′(t)‖ points in the direction the curve is bending. Both have length 1, and N is always perpendicular to T.
The arc length of a space curve r(t) = ⟨x(t), y(t), z(t)⟩ from t = a to t = b is L = ∫ ‖r′(t)‖ dt, where ‖r′(t)‖ = √(x′² + y′² + z′²) is the speed. You add up speed times time along the curve. It is the 2D arc length formula with one more term for z.
Vector functions and space curves go together: a vector function r(t) = ⟨f(t), g(t), h(t)⟩ assigns a 3D vector to each value of t, and the tips of those vectors trace a curve in space. You find limits, derivatives and integrals of r(t) one component at a time, and the derivative r′(t) points along the curve.