Consider the function y=x^2+5x-2 . What would happen to the graph if (x -1) was substituted in place of the x?
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Let
The curl is
where denotes the partial derivative operator with respect to
. Recall that
and that for any two vectors and
,
, and
.
The cross product reduces to
When you compute the partial derivatives, you'll find that all the components reduce to 0 and
which means is indeed conservative and we can find
.
Integrate both sides of
with respect to and
Differentiate both sides with respect to and
Now
and differentiating with respect to gives
for some constant . So