Question

If \( u(x, y, z) = x^2y + y^2z + z^2x \), the value of \[ \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} + \frac{\partial u}{\partial z} { at the point } (1,1,1) { is:} \]

Hint:

For multivariable functions, calculate partial derivatives with respect to each variable and then substitute the given values for the variables.

Answer 1

First, calculate the partial derivatives of \( u(x, y, z) \): \[ u(x, y, z) = x^2y + y^2z + z^2x \] Step 1: Calculate \( \frac{\partial u}{\partial x} \). \[ \frac{\partial u}{\partial x} = \frac{\partial}{\partial x} \left( x^2y + y^2z + z^2x \right) = 2xy + z^2 \] Step 2: Calculate \( \frac{\partial u}{\partial y} \). \[ \frac{\partial u}{\partial y} = \frac{\partial}{\partial y} \left( x^2y + y^2z + z^2x \right) = x^2 + 2yz \] Step 3: Calculate \( \frac{\partial u}{\partial z} \). \[ \frac{\partial u}{\partial z} = \frac{\partial}{\partial z} \left( x^2y + y^2z + z^2x \right) = y^2 + 2zx \] Now, evaluate these partial derivatives at the point \( (1, 1, 1) \): \[ \frac{\partial u}{\partial x} \bigg|_{(1,1,1)} = 2(1)(1) + (1)^2 = 2 + 1 = 3 \] \[ \frac{\partial u}{\partial y} \bigg|_{(1,1,1)} = (1)^2 + 2(1)(1) = 1 + 2 = 3 \] \[ \frac{\partial u}{\partial z} \bigg|_{(1,1,1)} = (1)^2 + 2(1)(1) = 1 + 2 = 3 \] Thus, the sum is: \[ \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} + \frac{\partial u}{\partial z} = 3 + 3 + 3 = 9 \] % Correct Answer Correct Answer:} \( 9 \)