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## Main Question or Discussion Point

say you have a function f(x,y)

[itex]\nabla[/itex]f= [itex]\partial[/itex]f/[itex]\partial[/itex]x + [itex]\partial[/itex]f/[itex]\partial[/itex]y

however when y is a function of x the situation is more complicated

first off [itex]\partial[/itex]f/[itex]\partial[/itex]x = [itex]\partial[/itex]f/[itex]\partial[/itex]x +([itex]\partial[/itex]f/[itex]\partial[/itex]y) ([itex]\partial[/itex]y/[itex]\partial[/itex]x)

( i wrote partial of y to x in case y was dependent on some other variable)

the [itex]\partial[/itex]f/[itex]\partial[/itex]x appears on both sides...what does this mean?do they can cancel? are their values equal?

my best guess is the partial with respect to x on the left side assumes non constant y, whereas the partial on the right wrt x assumes constant y.... how would you even show that in notation

now suppose we have a vector function

thanks

[itex]\nabla[/itex]f= [itex]\partial[/itex]f/[itex]\partial[/itex]x + [itex]\partial[/itex]f/[itex]\partial[/itex]y

however when y is a function of x the situation is more complicated

first off [itex]\partial[/itex]f/[itex]\partial[/itex]x = [itex]\partial[/itex]f/[itex]\partial[/itex]x +([itex]\partial[/itex]f/[itex]\partial[/itex]y) ([itex]\partial[/itex]y/[itex]\partial[/itex]x)

( i wrote partial of y to x in case y was dependent on some other variable)

the [itex]\partial[/itex]f/[itex]\partial[/itex]x appears on both sides...what does this mean?do they can cancel? are their values equal?

my best guess is the partial with respect to x on the left side assumes non constant y, whereas the partial on the right wrt x assumes constant y.... how would you even show that in notation

now suppose we have a vector function

**F**(x,y(x)), what is then the divergence of**F**, when we put in the operator [itex]\nabla[/itex] do we assume constant y or non constant y? and in which case does the divergence theorem hold?thanks