where f(x,y) is a function of two variables and vector (a,b) is a unit vector, we noticed how it is not always necessary to take the directional vector and unitize it only to plug in variable values based on what point we are taking the derivative at. We can use angles instead.
How does this work? When we originally used vectors, we had vectors such as:
unitized it into:
found the partial derivatives with respect to each variable, and multiplied it out to get the directional derivative, according to the above mentioned formula.
If we use angles, our vector now looks like:
and quite conveniently, any value of theta plugged into this 'angle vector' will also be a unit vector. This can be easily proven by finding the magnitude of u.
Another convenient property of this 'angle vector' is that no point is necessary. The angle value implies movement in a certain direction along both the x and y-axes.
Happy deriving!
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