The gradient ∇ϕ
abla\phi of the scalar field ϕ(r)=x2+2xy+x2z2 \phi(\mathbf{r}) = x^2+2xy+x^2 z^2 is equal to

Question

The gradient ∇ϕ 
abla\phi of the scalar field ϕ(r)=x2+2xy+x2z2 \phi(\mathbf{r}) = x^2+2xy+x^2 z^2 is equal to

BlackTom AI · Accepted Answer

The problem asks for the gradient ∇φ of the scalar field φ(r) = x^2 + 2xy + x^2 z^2.

First, compute the partial derivatives with respect to each coordinate:
- ∂φ/∂x: Differentiate x^2 with respect to x to get 2x; differentiate 2xy with respect to x to get 2y; differentiate x^2 z^2 with respect to x to get 2x z^2. Summing these gives ∂φ/∂x = 2x + 2y + 2x z^2 = 2(x + y + x z^2).
- ∂φ/∂y: Only the term 2xy depends on y, giving ∂φ/∂y = 2x.
- ∂φ/∂z: Only the term x^2 z^2 depends on z, giving ∂φ/∂z = 2 x^2 z.

Thus the gradient is ∇φ = [2(x + y + x z^2)] î_x + [2x] î_y + [2x^2 z] î_z.

Now evaluate each answer option against these components:
Option 1: 2(x + y + xz^2) e_x + 2y e_y + 2x^2 z e_z
- The x-component matches 2(x + y + x z^2). The y-component should be 2x, but this option has 2y, which is incorrect. The z-component 2x^2 z matches, but a single incorrect component makes the option incorrect overall.

Option 2: 2(x + y + xz) e_x + 2x e_y + 2x^2 z e_z
- The x-component is 2(x + y + x z), which uses xz instead of x z^2; this is inconsistent with the correct 2(x + y + x z^2). The y- and z-components are 2x and 2x^2 z, which are correct for those coordinates, but because the x-component is incorrect, this option is incorrect.

Option 3: 2(x + y + xz^2) e_x + 2x e_y + 2x^2 z e_z
- The x-component is 2(x + y + x z^2), which matches the correct ∂φ/∂x. The y-component is 2x, which matches ∂φ/∂y. The z-component is 2x^2 z, which matches ∂φ/∂z. All three components align with the calculated gradient, so this option is consistent with ∇φ.

Taking all options into account, Option 3 provides the correct gradient components, while Options 1 and 2 each contain at least one incorrect component that does not match the calculated partial derivatives. The gradient components are derived directly from differentiating φ with respect to x, y, and z, and these results align with the form in Option 3.

PHAS0009_24-25 1.3 Quiz 1 [0.25 hrs]

The gradient ∇ϕ \nabla\phi of the scalar field ϕ(r)=x2+2xy+x2z2 \phi(\mathbf{r}) = x^2+2xy+x^2 z^2 is equal to

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