Final answer:
The magnitude of the gradient of the scalar function f(x, y, z) = x²yz³ + e^y evaluated at point P(3, 0, 1) is 10.
Step-by-step explanation:
The magnitude (norm) of the gradient of a scalar function is calculated by taking the square root of the sum of the squared partial derivatives of the function. In this case, we have the function f(x, y, z) = x²yz³ + e^y.
To find the gradient at point P(3, 0, 1), we need to calculate the partial derivatives and evaluate them at that point. The partial derivatives of f with respect to x, y, and z are:
∂f/∂x = 2xyz³
∂f/∂y = x²z³ + e^y
∂f/∂z = 3x²yz²
Substituting the values x = 3, y = 0, z = 1 into these partial derivatives:
∂f/∂x = 2(3)(0)(1)³ = 0
∂f/∂y = (3)²(1)³ + e^0 = 9 + 1 = 10
∂f/∂z = 3(3)²(0)(1)² = 0
The magnitude of the gradient at point P is then calculated as:
|∇f(P)| = √(0)² + (10)² + (0)² = √100 = 101