Functions of several variables, partial derivatives, gradients, and multiple integrals — extending calculus to higher dimensions.
Partial Derivatives: One Variable at a Time
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When a function depends on multiple variables, we use partial derivatives to study how it changes with respect to each variable individually while holding the others constant. The partial derivative with respect to x, written with a curly d (the symbol for partial differentiation), measures the rate of change in the x-direction while treating all other variables as if they were constants. The gradient of a function is a vector formed by all its partial derivatives. It points in the direction of steepest ascent — the direction in which the function increases most rapidly. The magnitude of the gradient gives the rate of that steepest increase.
The gradient. The upside-down triangle is called "nabla" or "del." It converts a scalar function into a vector field.
Multiple Integrals: Volume and Beyond
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Just as a single integral computes area under a curve, a double integral computes volume under a surface. When you see two integral signs stacked together, it means you integrate over a two-dimensional region. The order of integration can often be swapped (Fubini's theorem) as long as the function is continuous. A triple integral integrates over a three-dimensional region to compute mass, charge, or any quantity distributed in space. The Jacobian determinant is crucial for changing variables — it corrects for the distortion caused by the coordinate transformation, similar to how substitution in single-variable calculus introduces a factor du over dx.
Double integral. dA represents the differential area element. The region R determines the limits of integration.
Optimization in Multiple Dimensions
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Finding maxima and minima of multivariable functions requires setting the gradient equal to zero — all partial derivatives must simultaneously vanish. The second derivative test uses the Hessian matrix (the matrix of all second partial derivatives) to classify critical points as local minima, local maxima, or saddle points. For problems with constraints, Lagrange multipliers provide an elegant method: introduce a new variable lambda and construct the Lagrangian function. Setting the gradient of the Lagrangian to zero yields the constrained optimum. This technique is fundamental to economics (utility maximization), physics (principle of least action), and machine learning (regularized optimization).
Lagrange multiplier condition. At a constrained optimum, the gradient of f is parallel to the gradient of the constraint g.
Key Facts
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Key Facts
The order of partial differentiation does not matter for smooth functions (Clairaut's theorem): the partial derivative with respect to x then y equals the partial derivative with respect to y then x. The Hessian matrix must be positive definite (all eigenvalues positive) for a local minimum, and negative definite for a local maximum.
Worked Example
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Worked Example
Find the critical points of f(x,y) = x² + y² − 2x − 4y + 5. Partial derivatives: fx = 2x−2, fy = 2y−4. Set to zero: x=1, y=2. Hessian: fxx=2, fyy=2, fxy=0. Determinant = 4 > 0 and fxx > 0, so (1,2) is a local minimum.
Surface Explorer
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Visualize how a surface changes as you modify parameters. Adjust the coefficient to see how the shape of a multivariable function responds.