We are given the vector field F = (x³, y³, z³) and the surface of a sphere defined by the equation x² + y² + z² = a². We need to find the flux of F across this closed surface.

To evaluate this triple integral over a sphere, it is most convenient to convert to spherical coordinates. In spherical coordinates, x² + y² + z² = r² and the volume element dV = r² sin dr d dθ. The limits of integration for a sphere of radius a are r from 0 to a, from 0 to π, and θ from 0 to 2π.

To evaluate this triple integral over a sphere, it is most convenient to convert to spherical coordinates. In spherical coordinates, x² + y² + z² = r² and the volume element dV = r² sin dr d dθ. The limits of integration for a sphere of radius a are r from 0 to a, from 0 to π, and θ from 0 to 2π.

Now we evaluate each of the separated integrals. The integral with respect to θ gives 2π. The integral with respect to gives 2. The integral with respect to r gives (a^5)/(5). Multiplying these results together with the constant 3 gives the total flux.

Now we evaluate each of the separated integrals. The integral with respect to θ gives 2π. The integral with respect to gives 2. The integral with respect to r gives (a^5)/(5). Multiplying these results together with the constant 3 gives the total flux.

Find the flux of F⃗ = (x³, y³, z³) across the surface of the sphere x² + y² + z² = a².

Answer: The flux of $\vec{F}$ across the surface of the sphere is $\frac{12\pi a^5}{5}$ .

Step-by-step solution

Step 1: Identify the vector field and surface

We are given the vector field $\vec{F} = (x^3, y^3, z^3)$ and the surface of a sphere defined by the equation $x^2 + y^2 + z^2 = a^2$ . We need to find the flux of $\vec{F}$ across this closed surface.

Step 2: Apply the Divergence Theorem

Since the surface is a closed sphere, we can use the Divergence Theorem (also known as Gauss's Theorem) to simplify the calculation. The theorem states that the flux of a vector field across a closed surface is equal to the triple integral of the divergence of the field over the volume enclosed by the surface.

Step 3: Calculate the divergence of the vector field

First, we need to compute the divergence of the vector field $\vec{F}$ . The divergence is the sum of the partial derivatives of each component with respect to its corresponding coordinate. This gives us $3x^2 + 3y^2 + 3z^2$ , which can be factored as $3(x^2 + y^2 + z^2)$ .

Step 4: Substitute divergence into the integral

Now we substitute the calculated divergence into the volume integral from the Divergence Theorem. The volume $V$ is the interior of the sphere $x^2 + y^2 + z^2 = a^2$ .

Step 5: Convert to spherical coordinates

To evaluate this triple integral over a sphere, it is most convenient to convert to spherical coordinates. In spherical coordinates, $x^2 + y^2 + z^2 = r^2$ and the volume element $dV = r^2 \sin\phi \, dr \, d\phi \, d\theta$ . The limits of integration for a sphere of radius $a$ are $r$ from $0$ to $a$ , $\phi$ from $0$ to $\pi$ , and $\theta$ from $0$ to $2\pi$ .

Step 6: Evaluate the integrals

Now we evaluate each of the separated integrals. The integral with respect to $\theta$ gives $2\pi$ . The integral with respect to $\phi$ gives $2$ . The integral with respect to $r$ gives $\frac{a^5}{5}$ . Multiplying these results together with the constant $3$ gives the total flux.

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