Find the gradient vector field of \( f(x, y, z)=\sqrt{3 x^{2}+y^{2}+2 z^{2}} \) \[ \square \]

To find the gradient vector field of the function \( f(x, y, z)=\sqrt{3 x^{2}+y^{2}+2 z^{2}} \), we need to compute the partial derivatives with respect to each variable \( x \), \( y \), and \( z \). The gradient vector \( \nabla f \) is given by: \[ \nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right). \] First, let's calculate each partial derivative: 1. **Partial derivative with respect to \( x \)**: \[ \frac{\partial f}{\partial x} = \frac{1}{2\sqrt{3 x^{2}+y^{2}+2 z^{2}}} \cdot (6x) = \frac{3x}{\sqrt{3x^{2}+y^{2}+2z^{2}}}. \] 2. **Partial derivative with respect to \( y \)**: \[ \frac{\partial f}{\partial y} = \frac{1}{2\sqrt{3 x^{2}+y^{2}+2 z^{2}}} \cdot (2y) = \frac{y}{\sqrt{3x^{2}+y^{2}+2z^{2}}}. \] 3. **Partial derivative with respect to \( z \)**: \[ \frac{\partial f}{\partial z} = \frac{1}{2\sqrt{3 x^{2}+y^{2}+2 z^{2}}} \cdot (4z) = \frac{2z}{\sqrt{3x^{2}+y^{2}+2z^{2}}}. \] Now, we can combine these into the gradient vector: \[ \nabla f = \left( \frac{3x}{\sqrt{3 x^{2}+y^{2}+2 z^{2}}}, \frac{y}{\sqrt{3 x^{2}+y^{2}+2 z^{2}}}, \frac{2z}{\sqrt{3 x^{2}+y^{2}+2 z^{2}}} \right). \] Thus, the gradient vector field of \( f(x, y, z)=\sqrt{3 x^{2}+y^{2}+2 z^{2}} \) is: \[ \nabla f = \left( \frac{3x}{\sqrt{3 x^{2}+y^{2}+2 z^{2}}}, \frac{y}{\sqrt{3 x^{2}+y^{2}+2 z^{2}}}, \frac{2z}{\sqrt{3 x^{2}+y^{2}+2 z^{2}}} \right). \]