If u=(x^2+y^2+z^2)^-1/2
find the integer value of k given that
(d^2u/dx^2)+(d^2/dy^2)+ (d^2u/dz^2)=k
To find the value of k, we first need to compute the second partial derivatives of u with respect to x, y, and z.
Given that u = (x^2 + y^2 + z^2)^(-1/2), we can simplify it using the chain rule as follows:
∂u/∂x = ∂u/∂(x^2) * ∂(x^2)/∂x = (-1/2)(x^2 + y^2 + z^2)^(-3/2) * 2x = -x(x^2 + y^2 + z^2)^(-3/2)
To find the second partial derivative, we differentiate again:
∂^2u/∂x^2 = ∂/∂x (-x(x^2 + y^2 + z^2)^(-3/2))
= -(x^2 + y^2 + z^2)^(-3/2) + (-1)(x)(-3/2)(x^2 + y^2 + z^2)^(-5/2)(2x)
= -(x^2 + y^2 + z^2)^(-3/2) + (3x^2(x^2 + y^2 + z^2)^(-5/2))
Using the same process, we can find the second partial derivatives with respect to y and z:
∂^2u/∂y^2 = -(y^2 + x^2 + z^2)^(-3/2) + (3y^2(y^2 + x^2 + z^2)^(-5/2))
∂^2u/∂z^2 = -(z^2 + x^2 + y^2)^(-3/2) + (3z^2(z^2 + x^2 + y^2)^(-5/2))
Finally, we can find the value of k by summing the three second partial derivatives:
k = ∂^2u/∂x^2 + ∂^2u/∂y^2 + ∂^2u/∂z^2
= -(x^2 + y^2 + z^2)^(-3/2) + (3x^2(x^2 + y^2 + z^2)^(-5/2))
- (y^2 + x^2 + z^2)^(-3/2) + (3y^2(y^2 + x^2 + z^2)^(-5/2))
- (z^2 + x^2 + y^2)^(-3/2) + (3z^2(z^2 + x^2 + y^2)^(-5/2))
By simplifying the expression above, we can find the integer value of k.