A gradient field (analysing a picture)

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SUMMARY

The discussion centers on the analysis of a gradient field represented in a picture, specifically examining how the function value f changes as one moves within the gradient. The participants confirm that moving slightly to the right from the point (1, -1) results in an increase in the function value due to the positive component of the gradient in that direction. The context involves understanding the relationship between gradient fields and level curves, particularly within a defined circular region.

PREREQUISITES
  • Understanding of gradient fields in multivariable calculus
  • Familiarity with level curves and their significance
  • Basic knowledge of function behavior in relation to gradients
  • Ability to interpret graphical representations of mathematical concepts
NEXT STEPS
  • Study the properties of gradient fields in multivariable calculus
  • Learn about the implications of level curves on function behavior
  • Explore the concept of directional derivatives and their calculations
  • Investigate the relationship between gradients and optimization problems
USEFUL FOR

Students and professionals in mathematics, physics, and engineering who are interested in understanding gradient fields and their applications in analyzing functions.

Poetria
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Homework Statement
The gradient of the function f.
R - the region inside and on the boundary of the circle.
You start at the point (1,-1) and move slightly to the right. How does the value of f change?
Relevant Equations
The function has its maximum at the point (0, -2)
Gradient-field.jpg
I think it is increasing as you move from one level curve to the other with bigger value. Am I right?
 
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Explain the relationship with the picture -- if any

What is the complete problem statement ?
 
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Well, you have the picture I have posted. And you should answer the question I have posted. The function isn't given.

"Here is a picture of the gradient of a function f . Let R denote the region inside and on the boundary of the circle."
"If you start at the point (1, -1) and move slightly to the right, how does the value of f change?
 
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I agree with your answer. The gradient has a component in your direction going to the right, so the function value is increasing slightly as you move right.
 
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FactChecker said:
I agree with your answer. The gradient has a component in your direction going to the right, so the function value is increasing slightly as you move right.
Great. :) Thank you very much. :)
 

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