Final answer:
To find the gradient of the scalar function V = 4x²y³z⁴, we compute the partial derivatives with respect to x, y, and z, which results in the gradient vector ∇V = <8xy³z⁴, 12x²y²z⁴, 16x²y³z³>.
Step-by-step explanation:
The student asked how to find the gradient of the scalar function V = 4x²y³z⁴. The gradient of a scalar function is a vector consisting of the partial derivatives of that function with respect to each variable. To find the gradient of V, we take the partial derivative of V with respect to each variable (x, y, z).
With respect to x:
∂V/∂x = 8xy³z⁴
With respect to y:
∂V/∂y = 12x²y²z⁴
With respect to z:
∂V/∂z = 16x²y³z³
Therefore, the gradient vector of V is
∇V = <8xy³z⁴, 12x²y²z⁴, 16x²y³z³>.