Multivariable Calculus
Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving several variables, rather than just one.
Multivariable calculus may be thought of as an elementary part of advanced calculus. For advanced calculus, see calculus on Euclidean space. The special case of calculus in three dimensional space is often called vector calculus.
Topics Involved In Notes
- Functions of several variables
- Limits
- Partial Derivatives
Functions of several variables
We are interested in functions f from R n to R m (or more generally from a subset D ⊂ R n to R m called the domain of the function). A function f assigns to each x ∈ R n a point y ∈ R m and we write
y = f(x)
The set of all such points y is the range of the function.
Each component of y = (y1, . . . , ym) is real-valued function of x ∈ R n
written yi = f i(x)
and the function can also be written as the collection of n functions
If we also write out the components of x = (x1, . . . , xn), then are function can be written as m functions of n variables each:
y1 =f1(x1, . . . , xn)y2 =f2(x1, . . . , xn)
. . .
ym =fm(x1, . . . , xn)
The graph of the function is all pairs (x, y) with y = f(x) . It is a subset of Rn+m
limits
Consider a function y = f(x) from Rn to R m (or possibly a subset of R n ). Let x 0 = (x 0 1 , . . . xnn ) be a point in R n and let y 0 = (y 0 1 , . . . , y0 m ) be a point in R m. We say that y 0 is the limit of f as x goes to x 0 , written