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 show that (βΟ)Β·r = nΟ, where Ο(x, y, z) = xn + yn + zn and n is a non-zero real constant, we need to find the gradient of Ο and evaluate (βΟ)Β·r.
First, let's find the gradient of Ο, denoted as βΟ. The gradient is a vector that contains the partial derivatives of the scalar field with respect to each variable (x, y, and z). Each component of the gradient will be a partial derivative of Ο.
βΟ = (βΟ/βx, βΟ/βy, βΟ/βz)
Finding the partial derivatives:
βΟ/βx = n*x^(n-1), βΟ/βy = n*y^(n-1), βΟ/βz = n*z^(n-1).
Therefore, the gradient of Ο is:
βΟ = (n*x^(n-1), n*y^(n-1), n*z^(n-1)).
Now, let's calculate the dot product of βΟ and the position vector r = (x, y, z).
(βΟ)Β·r = n*x^(n-1)*x + n*y^(n-1)*y + n*z^(n-1)*z.
Simplifying this expression:
(βΟ)Β·r = n*x^n + n*y^n + n*z^n.
Now, recall the definition of Ο(x, y, z) = xn + yn + zn. If we substitute this into the equation above, we get:
(βΟ)Β·r = nΟ.
Therefore, we have shown that (βΟ)Β·r = nΟ for the given scalar field Ο(x, y, z) = xn + yn + zn and a non-zero real constant n.