Understanding Continuity and the Jacobian Matrix in Multivariable Functions

In summary, this conversation discusses the concepts of continuity and Jacobian in relation to mappings and Taylor series expansions. In (a), the continuity for a mapping f: RN to RM is defined and its relationship to the metric (norm) is discussed. In (b), the Jacobian J of f is defined and the Taylor series expansion for f up to first degree at x = x0 is mentioned. In (c), the concept of y = f(x) \in RM and its components yj and |f(x)|j are introduced, and the Jacobian of f is further discussed in relation to the gradients of yj with respect to x.
  • #1
squenshl
479
4

Homework Statement


a) Let f: RN to RM. Define continuity for mapping f. How does this relate to the notion of metric (norm)?
b) Define the Jacobian J of f. Write Taylor series expansion (for f) up to first degree at x = x0. Explain the terms.
c) Let y = f(x) [itex]\in[/itex] RM and yj = |f(x)|j = sum from k = 1 to N of ajkxk. What is the Jacobian of f? How are the rows of the Jacobian related to the gradients of yj with respect to x?


Homework Equations


Taylor series


The Attempt at a Solution


I think I can do a but I am completely stuck on b and c. Any help please.
 
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  • #2


Okay, what answer did you give for (a)? And (b) just asks for the definition of "Jacobian". Isn't that given in your book?
 
  • #3


Any ideas on 5c?
 

Related to Understanding Continuity and the Jacobian Matrix in Multivariable Functions

What is continuity for mapping?

Continuity for mapping is a fundamental concept in mathematics and refers to the idea that small changes in the input of a function result in small changes in the output. It is often used to describe the behavior of a function as its input approaches a certain value.

How is continuity for mapping defined mathematically?

A mapping is continuous at a point if the limit of the function at that point exists and is equal to the value of the function at that point. In other words, the function must be defined and have a finite value at that point, and the limit of the function as the input approaches that point must exist and be equal to the value of the function at that point.

What does it mean for a function to be continuous on an interval?

A function is continuous on an interval if it is continuous at every point within that interval. This means that the function must be defined and have a finite value at every point in the interval, and the limit of the function as the input approaches each point must exist and be equal to the value of the function at that point.

What are the three types of continuity for mapping?

The three types of continuity for mapping are pointwise continuity, uniform continuity, and differentiability. Pointwise continuity refers to the continuity of a function at a specific point, while uniform continuity refers to the continuity of a function on an entire interval. Differentiability is a stronger form of continuity that requires the function to be smooth and have a defined derivative at every point in the interval.

Why is continuity important in mathematics and science?

Continuity is important because it allows us to make predictions and draw conclusions about the behavior of a function. It also allows us to define and study important concepts such as limits, derivatives, and integrals, which are essential in many areas of mathematics and science. Continuity is also crucial in applications such as physics and engineering, where it is used to model and analyze real-world systems.

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