Vector Calculus | Part IA, 2004

Explain what is meant by an exact differential. The three-dimensional vector field F\mathbf{F} is defined by

F=(exz3+3x2(eyez),ey(x3z3),3z2(exey)ezx3)\mathbf{F}=\left(e^{x} z^{3}+3 x^{2}\left(e^{y}-e^{z}\right), e^{y}\left(x^{3}-z^{3}\right), 3 z^{2}\left(e^{x}-e^{y}\right)-e^{z} x^{3}\right)

Find the most general function that has Fdx\mathbf{F} \cdot \mathbf{d} \mathbf{x} as its differential.

Hence show that the line integral

P1P2Fdx\int_{P_{1}}^{P_{2}} \mathbf{F} \cdot \mathbf{d} \mathbf{x}

along any path in R3\mathbb{R}^{3} between points P1=(0,a,0)P_{1}=(0, a, 0) and P2=(b,b,b)P_{2}=(b, b, b) vanishes for any values of aa and bb.

The two-dimensional vector field G\mathbf{G} is defined at all points in R2\mathbb{R}^{2} except (0,0)(0,0) by

G=(yx2+y2,xx2+y2)\mathbf{G}=\left(\frac{-y}{x^{2}+y^{2}}, \frac{x}{x^{2}+y^{2}}\right)

(G(\mathbf{G} is not defined at (0,0)(0,0).) Show that

CGdx=2π\oint_{C} \mathbf{G} \cdot \mathbf{d} \mathbf{x}=2 \pi

for any closed curve CC in R2\mathbb{R}^{2} that goes around (0,0)(0,0) anticlockwise precisely once without passing through (0,0)(0,0).

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