Elementary Calculus: Proof of Theorem 1

PROOF OF THEOREM 1

We must find a potential function for Pi + Qj.

Assume ∂P/∂y = ∂Q/∂x. Pick a point (a, b) in D, and let f(x₀, y₀) be the line integral of Pi + Qj on the rectangular curve C from (a, b) to (a, y₀) to (x₀, y₀) (Figure 13.3.7). Thus

By the Lemma,

Thus

and

(i) Let C have the parametric equations

x = g(t) y = h(t), c₁ ≤ t ≤ c₂.

Then

A = (g(c_l), h(c₁)) and B = (g(c₂), h(c₂)).

By the Chain Rule,

Then

A similar computation works for piecewise smooth curves. This proves (i). (ii) Define f(x, y) by

where A = (a, b), X = (x, y). Let C be the rectangular curve from (a, b) to (a, y) to (x, y). Then

We already showed in the proof of Theorem 1 that this function f(x, y) is a potential function for Pi + Qj. To complete the proof we note that the following are equivalent.

Home Vector Calculus Independence of Path Proof Of Theorem 1


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Proof of Theorem 1 Figure 13.3. PROOF OF THEOREM 1 We must find a potential function for Pi + Qj. Assume ∂P/∂y = ∂Q/∂x. Pick a point (a, b) in D, and let f(x₀, y₀) be the line integral of Pi + Qj on the rectangular curve C from (a, b) to (a, y₀) to (x₀, y₀) (Figure 13.3.7). Thus By the Lemma, Thus and (i) Let C have the parametric equations x = g(t) y = h(t), c₁ ≤ t ≤ c₂. Then A = (g(c_l), h(c₁)) and B = (g(c₂), h(c₂)). By the Chain Rule, Then A similar computation works for piecewise smooth curves. This proves (i). (ii) Define f(x, y) by where A = (a, b), X = (x, y). Let C be the rectangular curve from (a, b) to (a, y) to (x, y). Then We already showed in the proof of Theorem 1 that this function f(x, y) is a potential function for Pi + Qj. To complete the proof we note that the following are equivalent.
Home Vector Calculus Independence of Path Proof Of Theorem 1