Once functions depend on more than one input — height as a function of latitude and longitude, temperature in a room, profit from price and advertising — single-variable calculus is not enough. Multivariable calculus extends derivatives and integrals to several variables; vector calculus organizes those ideas when outputs or domains live in the plane or space.

Partial derivatives and the gradient

A partial derivative holds every variable but one fixed and differentiates with respect to the free one. For $z = f(x,y)$, $\partial f/\partial x$ asks how $z$ changes when $x$ moves and $y$ stays put. The gradient $\nabla f$ packages all first partials into a vector that points in the direction of steepest increase and whose magnitude is that steepest slope (under the usual Euclidean geometry).

Intuition. Level curves (or surfaces) of $f$ are perpendicular to $\nabla f$. Walking along a contour keeps $f$ constant; walking with the gradient changes $f$ fastest.

Linear approximation in several variables

Near a point, a differentiable function looks like its tangent plane (or hyperplane): $f(\mathbf{x}) \approx f(\mathbf{a}) + \nabla f(\mathbf{a})\cdot(\mathbf{x}-\mathbf{a})$. That is the multivariable cousin of $f(a)+f'(a)(x-a)$.

Vector fields and “rates with direction”

A vector field assigns a vector to each point — wind velocity, fluid flow, force. Divergence measures local expansion or compression; curl measures local rotation. Line and surface integrals ask how much a field “pushes along a path” or “flows through a surface.” The big theorems (Green, Stokes, divergence) relate those integrals to derivatives of the field — multivariable cousins of the Fundamental Theorem.

Pitfall. Notation is crowded. Distinguish carefully between a scalar field $f$, a vector field $\mathbf{F}$, and operators $\nabla$, $\nabla\cdot$, $\nabla\times$. Writing “grad,” “div,” and “curl” in words while you learn is often clearer than drowning in symbols.

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