Multivariable Calculus

Multivariable Calculus

1. Vector calculus It is an approach to differential geometry as a set of formulas and techniques to solve very useful problems for engineering and physics.

   • vector: it is represented by a line segment, oriented within the three-dimensional Euclidean space. It has 3 elements: module, direction and sense.
   • the dot product: it is an algebraic operation that takes two sequences of numbers of equal length and returns a single number.
   • cross product: it is a binary operation between two vectors in three-dimensional space. The result is a vector perpendicular to the multiplying vectors.
   • Applications
     • Partial derivative: it is the derivative with respect to each of these variables, keeping the others as constants.
     • double and triple integrals
       • theorems
         • Green: gives the relationship between a line integral around a simple closed curve C and a double integral
         • Stokes: Given a vector field, the theorem relates the integral of the rotational of a vector field on a surface, with the line integral of the vector field on the boundary of the surface.
     • It is used mainly for the calculation of areas, volumes of regions and solids in revolution
     • coordinate systems
       • the plane
         • rectangular
         • polar
       • space
         • cylindrical
         • spherical
         • rectangular