Question

Use the divergence theorem to show the following identities for scalar fields u(x) and v(x) a) Green's first identity: ?_S (u grad v) · n? dS = ?_V [u?²v + (grad u) · (grad v)] dV; b) Green's second identity: ?_S (u grad v - v grad u) · n? dS = ?_V (u?²v - v?²u) dV.

          Use the divergence theorem to show the following identities for scalar fields u(x) and v(x)

a) Green's first identity:
?_S (u grad v) · n? dS = ?_V [u?²v + (grad u) · (grad v)] dV;

b) Green's second identity:
?_S (u grad v - v grad u) · n? dS = ?_V (u?²v - v?²u) dV.

Use the divergence theorem to show the following identities for scalar fields u(x) and v(x)

a) Green's first identity:
?S (u grad v) · n? dS = ?V [u?²v + (grad u) · (grad v)] dV;

b) Green's second identity:
?S (u grad v - v grad u) · n? dS = ?V (u?²v - v?²u) dV.

Added by Ryan M.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sri K

02/17/2023

Step 1

To do this, we use the divergence theorem. Divergence theorem: f(ugradv)-ndS = f[uV*v+ (gradu) (gradv)JdV ; Now, we need to find the terms in the parentheses. We first need to find the divergence of u(x) and v(x). To do this, we use the divergence theorem. Show more…

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Use the divergence theorem to show the following identities for scalar fields u(x) and v(x): a) Green's first identity: ∫_S (u grad v) · n̂ dS = ∫_V [u∇²v + (grad u) · (grad v)] dV; b) Green's second identity: ∫_S (u grad v - v grad u) · n̂ dS = ∫_V (u∇²v - v∇²u) dV.