I was reading a section on vector fields and realized I am confused about how to take partials of vector quantities. If V(x,y)= y(adsbygoogle = window.adsbygoogle || []).push({}); i-xj, I don't understand why the [itex]\partialx[/itex]= y and the [itex]\partialy[/itex]= -x. The problem is showing why the previous equation is not a gradient vector field (because the second-order partials are not equal). 3 questions arise for me:

1) It seems to me that the only way you could obtain this answer for the [itex]\partialx[/itex] part of the gradient would be to only look at theivalue and ignore thejvalue. But why can you ignore thejcomponent? Is that because of the gradient's definition

2) Following off of the previous question, would[itex]\partialx[/itex] be something different (still using these vector components) if we were not interested in the gradient? And thus, is there something about taking gradients of vectors that can lead to an inequality among partials (because of only looking at certain components at certain times)?

3)And how is it that [itex]\partialx[/itex] yi= y? Doesiserve as "x," so that it is like taking the partial of yx? If so why is that?

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# Partial Derivatives of Vectors and Gradients

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