Multi-variable calculus of variation is a branch of mathematics that deals with finding optimal solutions for functions that depend on multiple variables. It involves finding the extreme values of functionals, which are mathematical expressions that take in functions as inputs and give a real number as an output.
In single-variable calculus of variation, the functionals only depend on a single variable. In contrast, multi-variable calculus of variation deals with functionals that depend on multiple variables. This means that the solutions to multi-variable problems are functions with multiple inputs, while the solutions to single-variable problems are functions with a single input.
Multi-variable calculus of variation has many applications in physics, engineering, and economics. It is used to find optimal paths for objects moving through space, optimal shapes for structures, and optimal solutions for problems involving multiple variables, such as optimization and control problems.
A strong foundation in single-variable calculus, linear algebra, and differential equations is necessary for understanding multi-variable calculus of variation. Familiarity with basic concepts of functional analysis, such as function spaces and the calculus of variations, is also helpful.
Some good references for studying multi-variable calculus of variation include "Calculus of Variations" by I. M. Gelfand and S. V. Fomin, "Introduction to the Calculus of Variations" by Bernard Dacorogna, and "Calculus of Variations and Optimal Control Theory" by Daniel Liberzon. Online resources such as lectures, notes, and exercises can also be found on websites like Khan Academy and MIT OpenCourseWare.