What is a small change of a function with 2 variables?

Question

I am reading a book and am confused about how this equation is founded. I would have thought that delta C would just be equal to the sum of the partial derivatives?
C is a function depending on v1 and v2.

Answer 1

The partial derivatives just tell you how fast the function is changing, it doesn't tell you what it changes TO. It would be like saying that I am currently moving at 100 meters per second. That tells you how fast I'm going, but it doesn't tell you how far I've moved yet. In this case, you'd get just $\frac{dp}{dt}$, so what we need is a change in time that we can apply that speed to in order to get my change in position.

When we have a multivariable function we in general can change among any of our independent variables, and we can do so independently, so we need to add up the contributions of each of those changes. Hence we still need those deltas - the changes in the respective variables.

When the deltas are "very large" the accuracy of the computation goes down, but that's just calculus - generally you need to assume that the deltas are so small as to be infinitessimal in order to be "accurate".

Answer 2

Suppose $C(v)= av_1 + b v_2 + d$. Then you have $C(v+\Delta v) = a (v_1+\Delta v_1) + b (v_1+\Delta v_1) +d = C(v) + a \Delta v_1 + b\Delta v_2$.

Since $a= {\partial C(v) \over \partial v_1}, b= {\partial C(v) \over \partial v_2}$ we have $C(v+\Delta v) = C(v) +{\partial C(v) \over \partial v_1} \Delta v_1 + {\partial C(v) \over \partial v_2} \Delta v_2 $.