Chapter 6: Q24MP (page 338)

$\iint r \times ndσ$ over the entire surface of the hemisphere $x^{2} + y^{2} + z^{2} = 9$ , $z \geq 0$
where $r = xi + yj + zk$ .

Short Answer

Expert verified

The solution is $\iint_{\partial T} r \times ndσ = 54 π$ .

Step by step solution

Given Information.

The given information is,

$x^{2} + y^{2} + z^{2} = 9$ , $z \geq 0$

Definition of Divergence Theorem.

The divergence theorem, often known as Gauss' theorem or Ostrogradsky's theorem, is a theorem that connects the flow of a vector field across a closed surface to the field's divergence in the volume enclosed.

Find the solution.

Use the divergence theorem $∭_{T} \nabla \times VdT = \iint_{\partial T} V \times ndσ$ , where $\partial T$ is the surface area that encloses the volume $T$ .

$\nabla \times r = \frac{\partial r_{x}}{\partial x} + \frac{\partial r_{y}}{\partial y} + \frac{\partial r_{z}}{\partial z}$

role="math" localid="1657354820813" $= 1 + 1 + 1$

$= 3$

Here, $\partial σ$ is the entire surface of the hemisphere of radius $3$ centered at $(0, 0, 0)$ , so $T$ is just the volume of the sphere.

$\iint_{\partial T} r \times ndσ = \iint_{\partial T} \nabla \times rdT$

$= 3 \iint_{T} d T$

$= 3 (\frac{1}{2}) (\frac{4}{3}) (π) (3)^{3}$

$= 54 π$

Hence, the solution is $\iint_{\partial T} r \times ndσ = 54 π$ .

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$\iint r \times ndσ$ over the entire surface of the hemisphere $x^{2} + y^{2} + z^{2} = 9$ , $z \geq 0$
where $r = xi + yj + zk$ .

Short Answer

Step by step solution

Given Information.

Definition of Divergence Theorem.

Find the solution.

One App. One Place for Learning.

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∬r×ndσover the entire surface of the hemispherex2+y2+z2=9,z≥0where r=xi+yj+zk.

Short Answer

Step by step solution

Given Information.

Definition of Divergence Theorem.

Find the solution.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Geometrical and Physical Optics

Turning Points in Physics

Electric Charge Field and Potential

Magnetism and Electromagnetic Induction

Rotational Dynamics

Torque and Rotational Motion

Study anywhere. Anytime. Across all devices.

$\iint r \times ndσ$ over the entire surface of the hemisphere $x^{2} + y^{2} + z^{2} = 9$ , $z \geq 0$
where $r = xi + yj + zk$ .