For a scaler field ฯ(x, y, z) = x^n+y^n+z^n, show that (๐ปฯ ).r = nฯ , where n is a non-zero real constant.
To prove that (๐ปฯ).r = nฯ, where ฯ(x, y, z) = x^n + y^n + z^n and n is a non-zero real constant, we need to find the divergence of ฯ and then verify the given equality.
Step 1: Find the divergence of ฯ
The divergence of a scalar field ฯ is given by the dot product of the gradient (๐ป) operator and ฯ. The gradient of ฯ is calculated by taking the partial derivative of ฯ with respect to each variable (x, y, and z). Let's compute the divergence:
(๐ปฯ) = (โ/โx, โ/โy, โ/โz) ยท (x^n + y^n + z^n)
Applying the dot product, we get:
(๐ปฯ) = (โ/โx)(x^n + y^n + z^n) + (โ/โy)(x^n + y^n + z^n) + (โ/โz)(x^n + y^n + z^n)
Calculating the partial derivatives:
(๐ปฯ) = nx^(n-1) + ny^(n-1) + nz^(n-1)
Step 2: Verify (๐ปฯ).r = nฯ
Now, we need to calculate (๐ปฯ).r and nฯ to check if they are equal. The vector r is given by (x, y, z).
(๐ปฯ).r = (nx^(n-1) + ny^(n-1) + nz^(n-1)).(x, y, z)
= nx^(n-1)x + ny^(n-1)y + nz^(n-1)z
= nx^n + ny^n + nz^n (since n โ 0)
nฯ = n(x^n + y^n + z^n)
Comparing (๐ปฯ).r and nฯ, we see that they are equal.
Therefore, we have shown that (๐ปฯ).r = nฯ for ฯ(x, y, z) = x^n + y^n + z^n, where n is a non-zero real constant.