Consider the combination

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Calculus of change and accumulation · Calculus of change and accumulation

Mathematical Methods for Physics, Ch Partial differentiation, chain rules and Jacobians

Problem

(a y^2 + 2x)\,\mathrm{d}x + 4xy\,\mathrm{d}y

, where

a

is a constant. ((a)) For which value of

a

is it the change

\mathrm{d}f

in some function

f(x,y)

? Find

f

for that value. ((b)) For that

a

, give the route sum of the combination from

(0,0)

(1,2)

without choosing a route. ((c)) Now take

a = 1

. Add up the combination along two routes from

(0,0)

(1,2)

: first along

y=0

(1,0)

and then up to

(1,2)

; and first up

x=0

(0,2)

and then along

y=2

(1,2)

. What do the two answers tell you?

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