Final answer:
To find ∇ϕ(1,2,0), we need to find the partial derivatives of the scalar field ϕ(x,y,z) with respect to x, y, and z, and evaluate them at the point (1,2,0). The partial derivatives are ∂ϕ/∂x = 3x², ∂ϕ/∂y = 3y²e⁴ʸᶻ, and ∂ϕ/∂z = 4y³e⁴ʸᶻ. Evaluating them at (1,2,0), we get ∇ϕ(1,2,0) = (3,0,0).
Step-by-step explanation:
To find ∇ϕ(1,2,0), we need to find the partial derivatives of the scalar field ϕ(x,y,z) with respect to x, y, and z, and evaluate them at the point (1,2,0).
First, let's find ∂ϕ/∂x. Taking the derivative of x³ with respect to x gives us 3x². Since y and z are constants, their derivatives are both 0.
Next, let's find ∂ϕ/∂y. Taking the derivative of y³e⁴ʸᶻ with respect to y gives us 3y²e⁴ʸᶻ. Since x and z are constants, their derivatives are both 0.
Finally, let's find ∂ϕ/∂z. Taking the derivative of y³e⁴ʸᶻ with respect to z gives us 4y³e⁴ʸᶻ. Since x and y are constants, their derivatives are both 0.
Therefore, ∇ϕ(1,2,0) = (3(1)², 3(2)²e⁴(1)(0), 4(2)³e⁴(2)(0)) = (3,0,0).