Question

Let F, =M, i+Nyj+Pk and F2 =Mzi + N2j+Pyk be differentiable vector fields and let a and b be arbitrary real constants. Verify the following identities. a. V. (aF7 +bF2)=aV.Fy+bV.F2 b. Vx(aF, +bF2) =aVxF, +bVxF2 c. V. (F, ⓇF_)=F2.VxF,-F7.VxFz a. Start by expressing af, +bF, in terms of My, Ng, P1, M2, Ny, and P2- V.(aF, + bFy)=v-[i+Di+(k] a a Use the definition of the divergence of a vector field, denoted div For V.F, to expand the right side of the equation above. Recall that V = i əxlay + k az a a V.aF, +bF2) = + Evaluate these partial derivatives. V.(aF, + bf2) = ( + + (List the terms in the same order as they appear in the original list.) This result can be rearranged and rewritten as follows. OM ON OP1 Әм, ON, OP2 V.(F4 +bF2)=a + b х dy dz dx ду dz + a + a + b + The first three terms in this expression can be written as by the The last three terms can be written as by the It follows that this expression can be written as aV.F+bV.F2, completing the proof.

Answer 1

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