Cross Product Continuity: Showing Definition is Satisfied

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Homework Help Overview

The discussion revolves around demonstrating the continuity of the cross product function in the context of vector calculus. The original poster attempts to apply the definition of continuity to show that the cross product is continuous, but encounters difficulties in finding an appropriate delta.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants discuss the definition of continuity and the relationship between the vectors involved. Questions arise regarding the expressions for |f(x)-f(y)| and |x-y|, with attempts to relate these to the properties of the cross product.

Discussion Status

Some participants are engaging in a detailed exploration of the definitions and properties relevant to the problem. There is an indication that guidance is being offered, particularly in clarifying the expressions involved, although the original poster claims to have figured out the issue.

Contextual Notes

There may be constraints related to the specific definitions and properties of continuity and the cross product that are being examined, as well as the requirement to adhere to formal definitions in the context of homework assignments.

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1. Show that the cross product is a continuous function

3.
I have tried to apply the definition of continuity: find a delta such that
|x-y|< delta implies |f(x)-f(y)|< epsilon
but I'm having trouble finding a delta that would take me to the conclusion.
 
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Perhaps you should show us what you've tried. What is |f(x)-f(y)| in this case?
 
In this case, |f(x)-f(y)| is a real number (the length of a vector in R^3).
A better question is what is |x-y|. x is one pair of vectors in R^3 and y is another pair of vectors in R^3. Then what is |x-y|?
I have tried relating |x-y| to the fact that |a x b|=|a||b|sin(theta) but can't get anything that seems to be on the right direction.
 
I figured it out.
 

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