For a scaler field Ļ(x, y, z) = x^n+y^n+z^n, show that (š»Ļ ).r = nĻ , where n is a non-zero real constant.
To show that (š»Ļ).r = nĻ for the scalar field Ļ(x, y, z) = x^n + y^n + z^n, where n is a non-zero real constant, we need to find the Laplacian of Ļ and then compute the dot product of the gradient of Ļ with the position vector r.
First, let's find the Laplacian of Ļ:
Laplacian of Ļ, (š»Ļ), is given by:
(š»Ļ) = (āĀ²Ļ/āxĀ²) + (āĀ²Ļ/āyĀ²) + (āĀ²Ļ/āzĀ²)
To find the second partial derivatives of Ļ, let's differentiate Ļ with respect to each variable separately:
āĻ/āx = n*x^(n-1)
āĀ²Ļ/āxĀ² = n(n-1)x^(n-2)
āĻ/āy = n*y^(n-1)
āĀ²Ļ/āyĀ² = n(n-1)y^(n-2)
āĻ/āz = n*z^(n-1)
āĀ²Ļ/āzĀ² = n(n-1)z^(n-2)
Now, let's add up the second partial derivatives:
(š»Ļ) = n(n-1)(x^(n-2) + y^(n-2) + z^(n-2))
Next, we calculate the gradient of Ļ, (āĻ):
āĻ = (āĻ/āx)i + (āĻ/āy)j + (āĻ/āz)k
= (n*x^(n-1))i + (n*y^(n-1))j + (n*z^(n-1))k
Finally, let's compute the dot product between (āĻ) and r:
(āĻ) . r = (n*x^(n-1))x + (n*y^(n-1))y + (n*z^(n-1))z
= n*x^n + n*y^n + n*z^n
= nĻ
So, we have shown that (š»Ļ).r = nĻ for the given scalar field.