Poll Results
No votes. Be the first one to vote.
Lost your password? Please enter your email address. You will receive a link and will create a new password via email.
Please briefly explain why you feel this question should be reported.
Please briefly explain why you feel this answer should be reported.
Please briefly explain why you feel this user should be reported.
To compute the divergence theorem for the vector field ( mathbf{D} = frac{5r^2}{4} mathbf{i} ) in spherical coordinates between ( r = 1 ) and ( r = 2 ), we follow these steps:
1. Convert the Vector Field to Spherical Coordinates:
In spherical coordinates, the relationship to Cartesian coordinates is given by:
– ( x = r sin theta cos phi )
– ( y = r sin theta sin phi )
– ( z = r cos theta )
The vector ( mathbf{i} ) represents the unit vector in the x-direction, so we express ( mathbf{D} ) as:
[
mathbf{D} = frac{5r^2}{4} mathbf{i} = frac{5r^2}{4} ( sin theta cos phi , sin theta sin phi , cos theta )
]
2. Compute the Divergence:
The divergence in spherical coordinates for a vector field ( mathbf{D} = (D_r, D_theta, D_phi) ) is given by:
[
nabla cdot mathbf{D} = frac{1}{r^2} frac{partial}{partial
Answer: c
Explanation: ∫∫ ( 5r2/4) . (r2 sin θ dθ dφ), which is the integral to be evaluated. Since it is double integral, we need to keep only two variables and one constant compulsorily. Evaluate it as two integrals keeping r = 1 for the first integral and r = 2 for the second integral, with φ = 0→2π and θ = 0→ π. The first integral value is 80π, whereas second integral gives -5π. On summing both integrals, we get 75π.