Problem:
For each of the following functions, find the derivative vector for those points where it is defined:
a.
c.
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a. I know that . As a result, then
So, if
By taking the partial derivatives, then
....
So, the gradient would be
However, I have a hard time understanding the gradient vector/derivative vector. I'm slightly confused because I thought was called the gradient and not a vector. If I can just understand this, then I can do (c) as well.
Thank you for your time.
You can remember the definition of del.
For ,
For ,
(that is the standart basis of the space)
So as you see here, del is taken as a vector.
Now we can derive the operators (for ).
Let be a scalar function.
What we did here was multiplying a vector (del) by a scalar (f).
Let F be a vector function such that .
We just applied dot product here.
And curl,