What is the application of Gauss theorem on a pyramid?

In summary, the conversation discusses using Gauss's theorem to calculate an integral on a closed boundary surface of a pyramid with given vertices and a given vector function. The summary also includes a discussion of finding the boundaries for the integral, as well as a correction for a mistake in the given answer. Finally, the conversation addresses what to do if the perpendicular vectors are oriented inward, and concludes with a confirmation of the corrected answer.
  • #1
mathmari
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Hey! :eek:

Using Gauss theorem I want to calculate $\iint_{\Sigma}f\cdot NdA$, where $\Sigma$ is the closed boundary surface of the pyramid with vertices $(0,0,0), (1,0,0), (0,1,0), (0,0,1), (1,1,0)$ and $f(x,y,z)=(x^2y, 3y^2z, 9xz^2)$ and the perpendicular vectors $N$ to the inside of the pyramid.

From Gauss theorem we have that $$ \iint_{\Sigma}f\cdot N \ dA=\iiint_{\Omega}\nabla\cdot f \ dV$$ We have that $$\nabla \cdot f=\frac{\partial{(x^2y)}}{\partial{x}}+\frac{\partial{(3y^2z)}}{\partial{y}}+\frac{\partial{(9xz^2)}}{\partial{z}}=2xy+6yz+18xz$$

Now we have to find the boundaries for the integral. We have the pyramid

View attachment 7634

So, do we have to find the equation of each line segment of the pyramid? (Wondering)
 

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  • #2
mathmari said:
Hey! :eek:

Using Gauss theorem I want to calculate $\iint_{\Sigma}f\cdot NdA$, where $\Sigma$ is the closed boundary surface of the pyramid with vertices $(0,0,0), (1,0,0), (0,1,0), (0,0,1), (1,1,0)$ and $f(x,y,z)=(x^2y, 3y^2z, 9xz^2)$ and the perpendicular vectors $N$ to the inside of the pyramid.

From Gauss theorem we have that $$ \iint_{\Sigma}f\cdot N \ dA=\iiint_{\Omega}\nabla\cdot f \ dV$$ We have that $$\nabla \cdot f=\frac{\partial{(x^2y)}}{\partial{x}}+\frac{\partial{(3y^2z)}}{\partial{y}}+\frac{\partial{(9xz^2)}}{\partial{z}}=2xy+6yz+18xz$$

Now we have to find the boundaries for the integral. We have the pyramid

So, do we have to find the equation of each line segment of the pyramid? (Wondering)

Hey mathmari! (Smile)

I think we can make it a little easier for ourselves.
From your picture we can see that x and y are both bounded to a square of which the size depends on z.
For $z=0$ we have $0\le x,y \le 1$.
For $z=\frac 12$ we have $0\le x,y \le \frac 12$.
For $z=1$ we have $0\le x,y \le 0$.

Can we deduce what the integral boundaries should be from that? (Wondering)
 
  • #3
I like Serena said:
I think we can make it a little easier for ourselves.
From your picture we can see that x and y are both bounded to a square of which the size depends on z.
For $z=0$ we have $0\le x,y \le 1$.
For $z=\frac 12$ we have $0\le x,y \le \frac 12$.
For $z=1$ we have $0\le x,y \le 0$.

Can we deduce what the integral boundaries should be from that? (Wondering)

So, we have that $0\leq x,y\leq z$ and $0\leq z\leq 1$, right?

Then we get the following:

\begin{align*}\iiint_{\Omega}(2xy+6yz+18xz) \ dV&=\int_0^1\int_0^z\int_0^z(2xy+6yz+18xz) \ dxdydz \\ & = \int_0^1\int_0^z\left [x^2y+6yzx+9x^2z\right ]_{x=0}^z \ dydz \\ & = \int_0^1\int_0^z\left (z^2y+6yz^2+9z^3\right )\ dydz \\ & = \int_0^1\int_0^z\left (7yz^2+9z^3\right )\ dydz \\ & = \int_0^1\left [\frac{7}{2}y^2z^2+9z^3y\right ]_{y=0}^z\ dz \\ & = \int_0^1\left (\frac{7}{2}z^4+9z^4\right )\ dz \\ & = \int_0^1\frac{25}{2}z^4\ dz \\ & = \frac{5}{2}\left [ z^5\right ]_0^1 \\ & = \frac{5}{2}\end{align*}

The given answer is $-1,1$. What have I done wrong? (Wondering)
 
  • #4
Shouldn't it be $0<x,y<1-z$? (Wondering)
 
  • #5
I like Serena said:
Shouldn't it be $0<x,y<1-z$? (Wondering)

Oh yes (Blush) Then we get the folowing:
\begin{align*}\iiint_{\Omega}&(2xy+6yz+18xz) \ dV=\int_0^1\int_0^{1-z}\int_0^{1-z}(2xy+6yz+18xz) \ dxdydz \\ & = \int_0^1\int_0^{1-z}\left [x^2y+6yzx+9x^2z\right ]_{x=0}^{1-z} \ dydz \\ & = \int_0^1\int_0^{1-z}\left [(1-z)^2y+6yz(1-z)+9(1-z)^2z\right ] \ dydz \\ & = \int_0^1\int_0^{1-z}\left [(1-2z+z^2)y+6yz(1-z)+9(1-2z+z^2)z\right ] \ dydz \\ & = \int_0^1\int_0^{1-z}\left [y-2zy+z^2y+6yz-6yz^2+9z-18z^2+9z^3\right ] \ dydz \\ & = \int_0^1\int_0^{1-z}\left [y+4yz-5yz^2+9z-18z^2+9z^3\right ] \ dydz \\ & = \int_0^1\left [\frac{y^2}{2}+2y^2z-\frac{5}{2}y^2z^2+9zy-18z^2y+9z^3y\right ]_{y=0}^{1-z} \ dz \\ & = \int_0^1\left [\frac{(1-z)^2}{2}+2(1-z)^2z-\frac{5}{2}(1-z)^2z^2+9z(1-z)-18z^2(1-z)+9z^3(1-z)\right ] \ dz \\ & = \int_0^1\left [\frac{1-2z+z^2}{2}+2(1-2z+z^2)z-\frac{5}{2}(1-2z+z^2)z^2+9z-9z^2-18z^2+18z^3+9z^3-9z^4\right ] \ dz \\ & =\int_0^1\left [\frac{1}{2}-z+\frac{z^2}{2}+2z-4z^2+2z^3-\frac{5}{2}z^2+5z^3-\frac{5}{2}z^4+9z-27z^2+27z^3-9z^4\right ] \ dz \\ & = \int_0^1\left [\frac{1}{2}-\frac{23}{2}z^4+10z-33z^2+34z^3\right ] \ dz \\ & = \left [\frac{1}{2}z-\frac{23}{10}z^5+5z^2-11z^3+\frac{34}{4}z^4\right ]_0^1 \\ & = \frac{1}{2}-\frac{23}{10}+5-11+\frac{34}{4}\\ & = \frac{7}{10}\end{align*}

Is everything correct? (Wondering)

Is at the given answer a typo or have I done something wrong? (Wondering)
 
  • #6
mathmari said:
Is everything correct?

Is at the given answer a typo or have I done something wrong?

Everything looks correct to me. (Nod)
And W|A confirms that your calculation of the integral is correct as well.
So I think there is indeed a mistake in the given answer. (Thinking)
 
  • #7
I like Serena said:
Everything looks correct to me. (Nod)
And W|A confirms that your calculation of the integral is correct as well.
So I think there is indeed a mistake in the given answer. (Thinking)

Great! Thank you! (Happy)
 
  • #8
What would we do if the perpendicular vectors $N$ would direct to the outside of the pyramid? (Wondering)
 
  • #9
mathmari said:
What would we do if the perpendicular vectors $N$ would direct to the outside of the pyramid?

Oh! I overlooked that before! (Wait)

Gauss's theorem assumes that the perpendicular vectors are oriented outward.
From wiki:
The closed manifold ∂V is quite generally the boundary of V oriented by outward-pointing normals, and n is the outward pointing unit normal field of the boundary ∂V.

Since the problem statement says that they are oriented inward, it means that we have to take the opposite. (Thinking)
 
  • #10
I like Serena said:
Oh! I overlooked that before! (Wait)
Gauss's theorem assumes that the perpendicular vectors are oriented outward.
Since the problem statement says that they are oriented inward, it means that we have to take the opposite.

So, we have to write a "-" everywhere and so we get $-0.7$, right? (Wondering)
 
  • #11
mathmari said:
So, we have to write a "-" everywhere and so we get $-0.7$, right?

Yep. (Nod)
 
  • #12
I like Serena said:
Yep. (Nod)

Thank you! (Sun)
 

FAQ: What is the application of Gauss theorem on a pyramid?

What is Gauss theorem on pyramid?

Gauss theorem on pyramid is a mathematical principle that states that the sum of the areas of the lateral faces of a pyramid is equal to half the product of the perimeter of the base and the slant height of the pyramid.

How is Gauss theorem on pyramid derived?

Gauss theorem on pyramid is derived from the concept of surface area and the Pythagorean theorem. It can also be proven using the Law of Cosines and the formula for the area of a triangle.

What is the significance of Gauss theorem on pyramid?

Gauss theorem on pyramid is important in geometry and engineering, as it allows for the calculation of the surface area of a pyramid using only the perimeter of its base and its slant height.

Can Gauss theorem on pyramid be applied to all types of pyramids?

Yes, Gauss theorem on pyramid can be applied to any type of pyramid, as long as it has a base that is a polygon and lateral faces that are triangles.

Are there any real-world applications of Gauss theorem on pyramid?

Yes, Gauss theorem on pyramid is commonly used in architecture and construction to calculate the surface area of pyramids, such as the Great Pyramid of Giza. It is also used in 3D modeling and computer graphics.

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