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