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In a vector field; Divergence of the gradient is
Question

In a vector field; Divergence of the gradient is

A.

curl

B.

unity

C.

zero

D.

Laplacian

Correct option is D

Div (grad ϕ)=(ϕ)=()ϕ=2ϕLet ϕ be a function of (x,y,z)Then grad (ϕ)=i^ϕx+j^ϕy+k^ϕzDivergence (grad ϕ)=2ϕx2+2ϕy2+2ϕz2\text{Div (grad } \phi) = \vec{\nabla} \cdot (\vec{\nabla} \phi) = (\vec{\nabla} \cdot \vec{\nabla}) \cdot \phi = \nabla^2 \phi \\[10pt]\text{Let } \phi \text{ be a function of } (x, y, z) \\[6pt]\text{Then grad } (\phi) = \hat{i} \frac{\partial \phi}{\partial x} + \hat{j} \frac{\partial \phi}{\partial y} + \hat{k} \frac{\partial \phi}{\partial z} \\[10pt]\text{Divergence (grad } \phi) = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial^2 \phi}{\partial z^2}​​

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