Vector field of gradient vector and contour plot

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SUMMARY

The discussion centers on the relationship between the gradient vector and contour plots of the scalar field defined by the function \( \frac{xy}{3} \). Participants confirm that the gradient vector is always perpendicular to the contour lines, which are hyperbolas in this case. Concerns were raised about the visibility of some level curves in the plot, suggesting a decrease in the sampling interval for better clarity. Overall, the gradient's orthogonality to the level curves is affirmed, despite some initial confusion regarding the representation of the scalar field.

PREREQUISITES
  • Understanding of gradient vectors in multivariable calculus
  • Familiarity with contour plots and level curves
  • Knowledge of scalar fields and their properties
  • Experience with plotting functions in a graphing tool
NEXT STEPS
  • Learn about gradient vector calculations in multivariable functions
  • Explore techniques for generating contour plots using software like MATLAB or Python's Matplotlib
  • Investigate the properties of hyperbolas and their applications in contour plotting
  • Study the effects of sampling intervals on the accuracy of contour representations
USEFUL FOR

Mathematicians, physics students, data scientists, and anyone involved in visualizing scalar fields and their gradients through contour plots.

Leo Liu
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Given the equation ##\frac{xy} 3##. It is a fact that the gradient vector function is always perpendicular to the contour graph of the original function. However it is not so evident in the plot above. Any thought will be appreciated.
 
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Leo Liu said:
However it is not so evident in the plot above.
Why do you say that? It looks true to me in the plot.
 
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FactChecker said:
Why do you say that? It looks true to me in the plot.
Some of the level curves are missing. Maybe I should decrease the sampling interval for the level curves?
 
The level curves are all hyperbolas. Try imagining a couple more of them and see if it's crosses through the gradients perpendicularly.
 
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Leo Liu said:
Given the equation ##\frac{xy} 3##. It is a fact that the gradient vector function is always perpendicular to the contour graph of the original function. However it is not so evident in the plot above. Any thought will be appreciated.
##xy/3## is a scalar field, it is not an equation. An equation requires an equal sign.

Apart from that, it is quite apparent from the plot that the gradient is orthogonal to all of the level curves you have provided. I do not understand what more you want from such a plot.
 
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