3. Green's first formula Suppose that f and g are scalar func- tions with continuous first- and second-order partial derivatives throughout a region D that is bounded by a closed piecewise smooth surface S. Show that /| f Vg•n do = /| (fV²g + Vf•Vg) dV. (10) Equation (10) is Green's first formula. (Hint: Apply the Diver- gence Theorem to the field F = f Vg.)

Calculus: Early Transcendentals
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Chapter1: Functions And Models
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3. Green's first formula Suppose that f and g are scalar func-
tions with continuous first- and second-order partial derivatives
throughout a region D that is bounded by a closed piecewise
smooth surface S. Show that
/|
f Vg•n do =
/| (fV²g + Vf•Vg) dV.
(10)
Equation (10) is Green's first formula. (Hint: Apply the Diver-
gence Theorem to the field F = f Vg.)
Transcribed Image Text:3. Green's first formula Suppose that f and g are scalar func- tions with continuous first- and second-order partial derivatives throughout a region D that is bounded by a closed piecewise smooth surface S. Show that /| f Vg•n do = /| (fV²g + Vf•Vg) dV. (10) Equation (10) is Green's first formula. (Hint: Apply the Diver- gence Theorem to the field F = f Vg.)
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