A vector field is given by v = x²i + 1 / 2y²j + 1 / 2 z²k. (a) Find div v. (b) Evaluate ∫V div v dV where V is the unit cube 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1.

Question

A vector field is given by [latex]\displaystyle\mathbf{v}=x^{2}\mathbf{i}+{\frac{1}{2}}y^{2}\mathbf{j}+{\frac{1}{2}}z^{2}\mathbf{k}.[/latex]

(a) Find div v.

(b) Evaluate [latex] extstyle{\int_{V}}[/latex] div v dV where V is the unit cube 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1.

Accepted Answer

(a) div v = 2x + y + z.
(b) We seek [latex]\int_{V}(2x+y+z)\,\mathrm{d}V[/latex] where V is the given unit cube. The small element of volume dV is equal to dz dy dx. With appropriate limits of integration the integral becomes

[latex]\int_{x=0}^{x=1}\int_{y=0}^{y=1}\ \int_{z=0}^{z=1}\ (2x+y+z)\,\mathrm{d}z \,\mathrm{d}y\,\mathrm{d}x[/latex]

This is a triple integral of the kind evaluated in Section 27.6.2:

[latex]\int_{x=0}^{x=1}\int_{y=0}^{y=1}\int_{z=0}^{z=1}(2x+y+z)\,\mathrm{d}z\,\mathrm{d}y\,\mathrm{d}x=\int_{x=0}^{x=1}\int_{y=0}^{y=1}\left[2x z+y z+{\frac{z^{2}}{2}}\right]_{0}^{1}\mathrm{d}y\,\mathrm{d}x\[/latex]

[latex]=\int_{x=0}^{x=1}\int_{y=0}^{y=1}\left(2x+y+{\frac{1}{2}}\right)\mathrm{d}y\,\mathrm{d}x\[/latex]

[latex]=\int_{x=0}^{x=1}\left[2x y+{\frac{y^{2}}{2}}+{\frac{y}{2}}\right]_{0}^{1}\mathrm{d}x\[/latex]

[latex]=\int_{x=0}^{x=1}{2x+1dx} \[/latex]

[latex]=[x^{2} + x ]^{1}_{0} [/latex]

= 2

Question 27.16: A vector field is given by v = x²i + 1 / 2y²j + 1 / 2 z²k. (......

Related Answered Questions

(a) Evaluate ∮C xy dx + x² dy around the sides of the square with vertices D(0, 0), E(1, 0), F(1, 1) and G(0, 1). (b) Convert the line integral to a double integral and verify Green’s theorem. ...

Verified Answer:

Evaluate ∮S v· dS where v = x²i + 1/2y²j + 1/2 z²k and where S is the surface of the unit cube 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1. The vector dS should be drawn on each of the six faces in an outward sense. ...

Verified Answer:

Verify the divergence theorem for the vector field v = x²i + 1 / 2y²j + 1 / 2 z²k over the unit cube 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1. ...

Verified Answer:

Verified Answer:

Evaluate ∫y = 0 ˆ y = 1 ∫x = 0 ˆ x = 2 – 2y 4x+5dx dy over the region described in Example 27.11. ...

Verified Answer:

Evaluate the double integral of f (x, y) = x² + 3xy over the region R indicated in Figure 27.13 by (a) integrating first with respect to x, and then with respect to y, (b) integrating first with respect to y, and then with respect to x. ...

Verified Answer:

Evaluate I = ∫x=0ˆx=1 ∫y=0ˆy=1 ∫z=0ˆz=1 x+y+zdzdydx ...

Verified Answer:

Verified Answer:

Show that the vector field F = ze^x sin yi + ze^x cos yj + e^x sin yk is a conservative field. ...

Verified Answer:

The vector field v is derivable from the potential Φ = 2xy + zx. Find v. ...

Verified Answer: