What Went Wrong in My Verification of the Divergence Theorem?

In summary, the conversation discusses the verification of the Divergence Theorem for a given surface and vector field. The attempt at a solution involves converting to polar coordinates and calculating an integral, but the final result does not match the expected answer. More information and clarification may be needed to determine the error in the calculation.
  • #1
khemist
248
0

Homework Statement



Let the surface, G, be the paraboloid [tex]z = x^2 + y^2[/tex] be capped by the disk [tex]x^2 + y^2 \leq 1[/tex] in the plane z = 1. Verify the Divergence Theorem for [tex]\textbf{F}(x,y,z) = 2x\textbf{i} - yz\textbf{j} + z^2\textbf{k}[/tex]

Homework Equations



I have solved the problem using the divergence theorem, that is no problem. However, I am having trouble verifying, where I used the formula [tex]\iint_G \textbf{F} \bullet d\textbf{S}[/tex]

The Attempt at a Solution



My projection on the xy-plane is a circle with the equation [tex]x^2 + y^2 = 1[/tex]. My n1 (normal vector), pointing from the plane z=1, is simply k. The n2, coming out of the surface [tex]z = x^2 + y^2[/tex], I got is 2x i + 2y j - k.

I converted my limits to polar coordinates, to get an integral

[tex]\int_0^{2\pi}\int_0^1 (4r^3(1+cos2\theta) - r^5(1-cos2\theta)) dr d\theta [/tex]

However, when I solve this, I get [tex]\frac{2\pi}{3}[/tex] when it should be [tex]\frac{5\pi}{2}[/tex].

Any ideas on what I have done wrong? thanks for the help
 
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  • #2
It's hard to say when you don't show intermediate steps. Your normals look fine.

I'll note that when I calculated the surface integral and the volume integral, I found them both equal to [itex]4\pi/3[/itex], not [itex]5\pi/2[/itex].
 

1. What is the Divergence Theorem?

The Divergence Theorem is a mathematical principle in vector calculus that relates the surface integral of a vector field to the volume integral of its divergence. It states that the flux of a vector field through a closed surface is equal to the volume integral of the divergence of the vector field over the region enclosed by the surface.

2. Why is the Divergence Theorem important?

The Divergence Theorem is important because it allows us to solve surface integrals over complicated surfaces by converting them into simpler volume integrals. It is a fundamental tool in many areas of physics and engineering, such as fluid dynamics, electromagnetism, and heat transfer.

3. How is the Divergence Theorem derived?

The Divergence Theorem is derived from Green's Theorem and the Fundamental Theorem of Calculus in two and three dimensions, respectively. It can also be derived from the Gauss's Law in electrostatics and the Gauss's Law for magnetism.

4. Can the Divergence Theorem be applied to any vector field?

No, the Divergence Theorem can only be applied to vector fields that are continuous and differentiable everywhere inside a closed surface. It may also be restricted by the properties of the surface itself, such as being smooth and having a well-defined orientation.

5. What are some real-world applications of the Divergence Theorem?

The Divergence Theorem has many practical applications, such as calculating the flow of fluids through pipes and channels, determining the electric and magnetic fields around a charged or magnetized object, and predicting the temperature distribution in a solid object. It is also used in computer graphics and computer simulations to model and visualize physical systems.

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