Editing Multivariable calculus (section)

==Introduction==
In single-variable calculus, operations like differentiation and integration are made to functions of a single variable. In multivariate calculus, it is required to generalize these to multiple variables, and the [[Domain of a function|domain]] is therefore multi-dimensional. Care is therefore required in these generalizations, because of two key differences between 1D and higher dimensional spaces:
# There are infinite ways to approach a single point in higher dimensions, as opposed to two (from the positive and negative direction) in 1D;
# There are multiple extended objects associated with the dimension; for example, for a 1D function, it must be represented as a curve on the 2D [[Cartesian plane]], but a function with two variables is a surface in 3D, while curves can also live in 3D space.

The consequence of the first difference is the difference in the definition of the limit and differentiation. Directional [[Limit of a function|limits]] and [[Directional derivative|derivative]]s define the limit and differential along a 1D parametrized curve, reducing the problem to the 1D case. Further higher-dimensional objects can be constructed from these operators.

The consequence of the second difference is the existence of multiple types of integration, including [[line integral]]s, [[surface integral]]s and [[volume integral]]s. Due to the non-uniqueness of these integrals, an [[antiderivative]] or [[indefinite integral]] cannot be properly defined.