please provide answer for this question Show that the vector field by ⃗= (x^2+4yz)̂+ (y^2+4zx)̂+ (z^2+4xy) ̂ is irrotational. Hence find the scalar potential.

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please provide answer for this question  Show that the vector field  by  ⃗= (x^2+4yz)̂+ (y^2+4zx)̂+ (z^2+4xy) ̂ is irrotational. Hence find the scalar potential.

nick1337 · Accepted Answer

We are saked to show that the vector field f→=(x2+4yz)i→+(y2+4zx)j→+(z2+4xy)k→ is irrational. Also we are asked to find scalar potential. Since we know that a vector filed A→ is irrational if curlA→=0→ or ∇→×A→=0→ Also with respect to a irrational field A→ there exist a scalar potential ⊘ such that A→=∇→⊘ Now solving for proving f→ irrational we will find curlf→ curlf→=∇→×f→=[i→j→k→∂∂x∂∂y∂∂zfxfyfz] Where ∂∂x,∂∂y and ∂∂z are partial derivatives with respect to and f→=fxi→+fyi→+fzk→ x,y and z respectively. Now ∇→×f→=[i→j→k→∂∂x∂∂y∂∂zx2+4yzy2+4zxz2+4xy] =i→[∂∂y(z2+4xy)−∂∂z(y2+4zx)]−j→[∂∂x(z2+4xy)−∂∂z(x2+4yz)]+k→[∂∂x(y2+4zx)−∂∂y(x2+4yz)] =i→(4x−4x)−j→(4y−4y)+k→(4z−4z) =i→(0)−j→(0)+k→(0) ∇→×f→=0→ Since ∇→×f→=0→ so f→ is irrational vector field. Now for scalar potential ⊘ since f→=∇⊘=(i→∂∂x+j→∂∂y+k→∂∂z)⊘ f→=∂⊘∂xi→+∂⊘∂yj→+∂⊘∂zk→ (x2+4yz)i→+(y2+4zx)j→+(z2+4xy)k→=∂⊘∂xi→+∂⊘∂yj→+∂⊘∂z

please provide answer for this question Show that the

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