Calculating Work and Flux Using Green's Theorem: A Square Example

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In summary: Thank you! (Smirk)In summary, we used Green's Theorem to calculate the work and flux for a closed anti-clockwise path around a square with boundaries at $x=0$, $x=1$, $y=0$, and $y=1$. The vector field was $\overrightarrow{F}=2xy\hat{\imath}+3x^2y\hat{ \jmath }$ and we determined the work to be $\frac{1}{2}$ and the flux to be $2$. We also noted that for the calculation of work, we used $\hat{n}=\hat{k}$ since the square lies in the $xy$ plane.
  • #1
mathmari
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MHB
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Hey! :eek:

"Use the Green's Theorem to calculate the work and the flux for the closed anti-clockwise direction that consists of the square which is determined by the lines $x=0$, $x=1$, $y=0$ and $y=1$ if $\overrightarrow{F}=2xy\hat{\imath}+3x^2y\hat{ \jmath }$."I have done the following:

$$\text{Work }: \oint_C{\overrightarrow{F}}dR=\int \int_R {\nabla \times \overrightarrow{F} \cdot \hat{n}}dA=\iint_R {\nabla \times \overrightarrow{F} \cdot \hat{k}}dA=\iint_R {(6xy-2x)}dA=\int_0^1 \int_0^1(6xy-2x)dxdy=\int_0^1(3y-1)dy=\frac{1}{2}$$

$$\text{Flux }: \oint_C{\overrightarrow{F} \cdot \hat{n}}ds=\iint_R{\nabla \cdot \overrightarrow{F}}dA=\iint_R{(2y+3x^2)}dA=\int_0^1 \int_0^1{(2y+3x^2)}dxdy=\int_0^1{(2y+1)}dy=2$$

Is this correct?
 
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  • #2
mathmari said:
Hey! :eek:

"Use the Green's Theorem to calculate the work and the flux for the closed anti-clockwise direction that consists of the square which is determined by the lines $x=0$, $x=1$, $y=0$ and $y=1$ if $\overrightarrow{F}=2xy\hat{\imath}+3x^2y\hat{ \jmath }$."I have done the following:

$$\text{Work }: \oint_C{\overrightarrow{F}}dR=\int \int_R {\nabla \times \overrightarrow{F} \cdot \hat{n}}dA=\iint_R {\nabla \times \overrightarrow{F} \cdot \hat{k}}dA=\iint_R {(6xy-2x)}dA=\int_0^1 \int_0^1(6xy-2x)dxdy=\int_0^1(3y-1)dy=\frac{1}{2}$$

$$\text{Flux }: \oint_C{\overrightarrow{F} \cdot \hat{n}}ds=\iint_R{\nabla \cdot \overrightarrow{F}}dA=\iint_R{(2y+3x^2)}dA=\int_0^1 \int_0^1{(2y+3x^2)}dxdy=\int_0^1{(2y+1)}dy=2$$

Is this correct?

Yep. All correct! (Happy)Btw, I would write \(\displaystyle \oint_C{\overrightarrow{F}} \cdot d\vec R\) or \(\displaystyle \oint_C{\overrightarrow{F}} \cdot \hat c dR\) to be notationally correct.
 
  • #3
I like Serena said:
Yep. All correct! (Happy)

Great! (Yes)

I like Serena said:
Btw, I would write \(\displaystyle \oint_C{\overrightarrow{F}} \cdot d\vec R\) or \(\displaystyle \oint_C{\overrightarrow{F}} \cdot \hat c dR\) to be notationally correct.

Ok! Thank you! (Smirk)
 
  • #4
At the calculation of work, we took $\hat{n}=\hat{k}$, because the square is at the plane $xy$, right?
 
  • #5
mathmari said:
At the calculation of work, we took $\hat{n}=\hat{k}$, because the square is at the plane $xy$, right?

Yes.
 
  • #6
I like Serena said:
Yes.

Ok!
 

FAQ: Calculating Work and Flux Using Green's Theorem: A Square Example

What is Green's Theorem?

Green's Theorem is a mathematical theorem that relates the work done by a force field on a closed curve in two dimensions to the double integral of the flux of the field over the region enclosed by the curve.

What is the significance of Green's Theorem?

Green's Theorem is significant because it allows for the calculation of the work done by a force field without having to evaluate line integrals. It also provides a useful tool for solving problems in physics, engineering, and other fields that involve vector fields.

How is Green's Theorem related to Stokes' Theorem?

Green's Theorem is a special case of Stokes' Theorem, which is a more general theorem that relates the line integral of a vector field along a curve to the surface integral of the curl of the field over the surface enclosed by the curve. Green's Theorem applies specifically to two-dimensional vector fields, while Stokes' Theorem applies to three-dimensional vector fields.

What is the difference between work and flux in Green's Theorem?

In Green's Theorem, work refers to the physical concept of force doing work on an object, while flux refers to the flow of a vector field through a surface. In the context of Green's Theorem, work is the line integral of a force field along a closed curve, while flux is the double integral of the same force field over the region enclosed by the curve.

What are some real-world applications of Green's Theorem?

Green's Theorem has many applications in physics, engineering, and other fields. It can be used to calculate the work done by a fluid moving through a pipe, the flow of electricity through a circuit, and the heat transfer in a material. It also has applications in vector calculus, differential equations, and computer graphics.

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