Multivariable Analysis

Multivariable analysis, also known as multivariate analysis, is a branch of mathematics that studies functions of more than one variable and their derivatives and integrals.

Partial Derivatives

In multivariable calculus, partial derivatives are used to measure how a function changes when only one of the variables is varied. The partial derivative of a function f(x, y, ..., z) with respect to x is denoted as ∂f/∂x and it measures the rate of change of f as x changes, keeping all other variables constant.

Multiple Integrals

Multiple integrals extend the concept of a definite integral to functions of more than one variable. For instance, a double integral allows for the calculation of the volume under a surface in three-dimensional space.

Gradient, Divergence, and Curl

The gradient of a scalar function gives a vector field that points in the direction of greatest rate of increase of the function. The divergence of a vector field is a scalar function that measures the degree to which the vector field is diverging at each point. The curl of a vector field gives a new vector field that measures the rotation of the original vector field.

Line and Surface Integrals

Line integrals are used to integrate a function along a curve in two or more dimensions. Surface integrals extend this concept to integrating over a surface in three or more dimensions.

Theorems of Multivariable Calculus

Multivariable calculus also includes several important theorems such as Green's theorem, Stokes' theorem, and the divergence theorem. These theorems establish important relationships between differential and integral calculus in higher dimensions.

Multivariable analysis has applications in many areas of physics, engineering, and computer science, among others.