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...As an example consider the vector eld e x y kq endpoints and you will nd that for any piecewise smooth curve c joining to z dx has same value namely zero exercise show line integral of equals each curves where is quarter circle denition let f be a continuous on connected open set in rn we say path independent u if given two points all join gives hint r course cannot prove by calculating its dierent pairs because there are innitely many possibilities so important question how can tell whether or not out us guided one most results elementary calculus rst fundamental theorem first integrals recall interval then new function g dened xf t dt b such this result ftc q extent yes provided joins denote case dene scalar analogy with now state subset whose specied point proof simplicity give have components key idea since free make special choice i choose custom designed figure shows need suitable parametrizations using these equations yf it follows from riemann see treating constant similarly ge...