The proof of the ftoli (fundamental theorem of line integrals) used which is chain rule for multivariable functions
What is Chain rule for multivariable function?
According to the chain rule, the derivative of f(g(x)) is f'(g(x))g' (x). In other words, it aids in the distinction of "composite functions." For instance, the fact that sin(x2) can be created as f(g(x)) when f(x)=sin(x) and g(x)=x2 indicates that this function is composite.
= [tex]\int\limits^a_c {f} \, dr[/tex] = [tex]\int\limits^b_a {f} \,[/tex] (r(t) . r (t)dt
= [tex]\int\limits^b_a[/tex](df/dn . dn/dt + df/dy . dy/dt +df/dz . dz/dt) = dt
by chain rule = [tex]\int\limits^b_a[/tex] d/dt [ [fr (t) ] dt]
= [tex]\int\limits^a_c[/tex] f.dr = f(r(b) - f(r)(a))
Chain rule for multivariable functions is used in the demonstration of the ftoli (fundamental theorem of line integrals).
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