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A Vector field F is given by ˉF=(x2−yz)ˆi+(y2−zx)ˆj+(z2−xy)ˆk
is irroational and hence find scalar point function ? such that F = ? ?
1 Answer
written 3.9 years ago by |
Part 1:F=(x2−yz)ˆi+(y2−zx)ˆj+(z2−xy)ˆk ∴r=xi+yj+zkCurl F=|ˆiˆjˆk∂∂x∂∂y∂∂zx2−yzy2−zxz2−xy|
=0−0+0=0∴F is a conservative field.∴there exists a scalar potential of F such that F=∇φ∴(x2−yz)ˆi+(y2−zx)ˆj+(z2−xy)ˆk=i∂φ∂x+j∂φ∂x+∂φ∂x
Comparing both sides gives,∂φ∂x=x2−yz;∂φ∂x=y2−zx;∂φ∂x=z2−xy
Nowdφ=∂φ∂xdx+∂φ∂ydy+∂φ∂zdz∴ dφ=(x2−yz)dx+(y2−zx)dy+(z2−xy)dzon Integration,φ=x33−yz.x+y33+z33+C∴Scalar potential of F=φ=13(x3+y3+z3)−xyz+C