What is the Derivative of Functions from R2->R2 and How is it a Linear Map?

  • Thread starter domhal
  • Start date
  • Tags
    Calculus
In summary, the conversation discusses the concept of differentiation of functions from R2 to R2, which is briefly mentioned in a complex analysis course. The book considers an example function f(x,y) = (x,-y) and explains that its derivative at a point is a linear map represented by its Jacobian matrix. The derivative of a function is a linear function that best approximates the function at that point. It is recommended to read the Bartle Elements of Real Analysis for a more thorough understanding of differentiation of such functions.
  • #1
domhal
4
0
I would like to learn about differentiation of functions from R2->R2. Such functions are mentioned briefly at the beginning of the text for my complex analysis course for sake of comparison. However, I find that I don't know very much about them.

The book considers f(x,y) = (x, -y), saying that f is differentiable and that "Its derivative at a point is the linear map given by its Jacobian...". I don't understand, for example, how the derivative is a linear map. I associate (perhaps incorrectly) the derivative with a rate of change. Could someone recommend some resources (books, websites etc) for understanding the differentiation of such functions?

Domhal
 
Physics news on Phys.org
  • #2
Think about this definition of the derivative, implicitly:
[tex] \lim_{h \rightarrow 0} \left ( f(\mathbf{x + h}) - f(\mathbf{x}) = \mathbf{h} Df(x) \right ) [/tex]
In this case, if [tex] \mathbf{h} \in R^2 [/tex] and [tex] f(\mathbf{x}) :R^2 \rightarrow R^2 [/tex] then, for the dimensionality to make sense, [tex]Df(\mathbf{x}) [/tex] has to be a 2x2 matrix.

I strongly recommend the Bartle Elements of Real Analysis to read more on this. It's chapters on many variable functions are quite interesting and thorough, as is the rest of the book.
 
  • #3
A more general definition of the derivative of any function (at a point) is the linear function that best approximates the function at that point.

In that sense, the derivative of, say, y= x2, at x= 1 is not the number 2 but the linear function y= 2x.

The derivative of f(x,y)= (x,-y) is the linear function whose matrix (in the i, j basis) is the matrix with columns (1,0), (0, -1).
 

What is R2->R2 Calculus?

R2->R2 Calculus, also known as multivariable calculus, is a branch of mathematics that deals with functions of several variables. It extends the concepts and techniques of single-variable calculus to functions of two or more variables.

What are the applications of R2->R2 Calculus?

R2->R2 Calculus has many practical applications in science and engineering, including physics, economics, and computer graphics. It is used to model and analyze systems with multiple variables, such as motion in three-dimensional space, optimization problems, and the behavior of fluids.

What are some key concepts in R2->R2 Calculus?

Some key concepts in R2->R2 Calculus include partial derivatives, multiple integrals, and vector calculus. Partial derivatives are used to find the rate of change of a function with respect to one variable while holding others constant. Multiple integrals are used to calculate volumes and surface areas of regions in two or more dimensions. Vector calculus involves vector-valued functions and their derivatives, and is used to describe and analyze the motion of objects in space.

What are the prerequisites for learning R2->R2 Calculus?

A strong foundation in single-variable calculus is essential for understanding R2->R2 Calculus. It is also helpful to have a working knowledge of algebra, trigonometry, and geometry. Familiarity with basic concepts in linear algebra and vectors is also beneficial.

What are some tips for succeeding in R2->R2 Calculus?

Some tips for succeeding in R2->R2 Calculus include practicing regularly, seeking help when needed, and understanding the fundamental concepts rather than just memorizing formulas. It is also important to have a strong grasp of single-variable calculus before moving on to multivariable calculus. Developing good problem-solving skills and using visualization techniques can also be helpful in understanding and applying the concepts of R2->R2 Calculus.

Similar threads

Replies
14
Views
1K
Replies
3
Views
1K
Replies
5
Views
974
  • Calculus
Replies
9
Views
1K
Replies
36
Views
3K
Replies
3
Views
1K
Replies
11
Views
2K
  • Calculus
Replies
4
Views
2K
Replies
1
Views
811
Replies
22
Views
2K
Back
Top