1. ## Help with Gradient Of function problem

Suppose (x,y) is a point in the first quadrant (x,y>0). If delta is the angle of (x,y) in polar representation, determine the length and direction of gradient of delta(x,y)?
can someone show me how to approach this type of question?
Thanks for the help!

2. ## Re: Help with Gradient Of function problem

Originally Posted by Ricebunny
the angle of (x,y) in polar representation
... equals arctan (y/x).

arctan(y/x) - Wolfram|Alpha

Notice the partial derivative wrt x. Click on 'show steps'. Just in case a picture helps...

... where (key in spoiler) ...

Spoiler:

... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to the main variable (in this case x), and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).

The general drift is...

... equals a vector composed of both partial derivatives:

Gradient - Wikipedia, the free encyclopedia

determine the length and direction
.. as for any vector: the length or magnitude using pythagoras and the direction relative to the x-axis using arctan as we did above for delta.

For length, simplify $\displaystyle \sqrt{{(\tfrac{-y}{x^2 + y^2})}^2 + {(\tfrac{x}{x^2 + y^2})}^2}$

For direction, arctan of a simplification of the 'y' component over the 'x'.

_________________________________________

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Balloon Calculus; standard integrals, derivatives and methods

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3. ## Re: Help with Gradient Of function problem

Thank you so much for the help!
Now I understand much better.