Question 26.2: If Φ = Φ (x, y, z) = 4x³y sin z, find ∇Φ....

Engineering Mathematics A foundation for Electronic, Electrical, Communications and Systems Engineers [1040491]

If $\phi=\phi(x,y,z)=4x^{3}y\,\mathrm{sin}\,z,$ , find ∇Φ.

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\phi(x,y,z)=4x^{3}y\sin z

so that by partial differentiation we obtain

{\frac{\partial\phi}{\partial x}}=12x^{2}y\sin z\\

{\frac{\partial\phi}{\partial y}}=4x^{3}\sin z\\

{\frac{\partial\phi}{\partial z}}=4x^{3}y\cos z

Therefore

\nabla\phi=12x^{2}y\sin z\mathbf{i}+4x^{3}\sin z\mathbf{j}+4x^{3}y\cos z\mathbf{k}

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