1 Chalkboard
Figure 1: What is the Green’s Theorem for flux
Figure 2: Green’s Theorem for flux
Figure 3: Green’s Theorem in normal form vs tangential form
Figure 4: Proof of Green’s Theorem for flux
Figure 5: Changing name of \(\vb{F}\) components
Figure 6: Example of Green’s Theorem for flux
2 Explain why flux is a line integral
2.1 Front
Explain why flux is a line integral
Let \(\vb{F}\) a vector field, and any curve
2.2 Back
We can divide up the curve and apply to each of the approximating line segments, so:
mass-transport rate across line segment \(\approx (\vb{F_k} \cdot \vu{n_k}) \Delta s_{k}\)
Add up all these subdivision and
mass-transport rate across \({\displaystyle C = \int_C \vb{F} \cdot \vb{n} \dd{s}}\)
So, this line integral is called flux of \(\vb{F}\) across
3 How can you write \(\vu{n}\dd{s}\) as vector?
3.1 Front
How can you write $\vu{n}\dd{s}$ as vector?
Write in terms of components, similarly to \(\dd{\vb{r}} = \ev{\dd{x}, \dd{y}}\)
3.2 Back
\(\vu{n}\) is 90 degrees clockwise to tangent vector \(\vb{T}\). As you can write \(\vb{T} \dd{s}\)
\(\dd{\vb{r}} = \vb{T} \dd{s} = \ev{\frac{\dd{x}}{\dd{s}}, \frac{\dd{y}}{\dd{s}}} \dd{s} = \ev{\dd{x}, \dd{y}}\)
Clockwise this vector 90 degrees
\(\vu{n} \dd{s} = \ev{\frac{\dd{y}}{\dd{s}}, - \frac{\dd{x}}{\dd{s}}} \dd{s} = \ev{\dd{y}, - \dd{x}}\)
4 What is the Green’s Theorem for flux?
4.1 Front
What is the Green’s Theorem for flux?
Let \(\vb{F} = \ev{M,N}\) a 2 dimensional flow field, and a simple closed curve, positively oriented
4.2 Back
- Flux of \(\vb{F}\) across \({\displaystyle = \oint_C M \dd{y} - N \dd{x}}\)
As \(\vu{n}\) points outwards, away from , the flux is positive where the flow is out of , flow into counts as negative flux
Applying Green’s Theorem to the line integral
\({\displaystyle \oint_C M \dd{y} - N \dd{x} = \oint_C - N \dd{x} + M \dd{y} = \iint_R (M_x - (- N_y)) \dd{A} = \iint_R (M_x + N_y) \dd{A}}\)
5 How is the flow if the flux of \(\vb{F}\) across is positive?
5.1 Front
How is the flow if the flux of $\vb{F}$ across $C$ is positive?
Let \(\vb{F} = \ev{M,N}\) a 2 dimensional flow field, and a simple closed curve, positively oriented
5.2 Back
It’s away from
As \(\vu{n}\) points outwards, away from , the flux is positive where the flow is out of , flow into counts as negative flux
6 What is the two-dimensional divergence of \(\vb{F}\)
6.1 Front
What is the two-dimensional divergence of $\vb{F}$
Let \(\vb{F} = \ev{M,N}\)
6.2 Back
\({\displaystyle \text{div} \vb{F} = M_x + N_y}\)
It’s a scalar function of two variables
7 Which is the physical interpretation of \(\text{div} \vb{F}\)?
7.1 Front
Which is the physical interpretation of $\text{div} \vb{F}$?
Let \(\vb{F}\) a flow field continuously differentiable
7.2 Back
If \(\vb{F}\) is continuously differentiable, then \(\text{div} \vb{F}\) is a continuous function. Looking at a small rectangle, the flux of \(\vb{F}\) across the sides of this rectangle is constant
For \({\displaystyle \iint_R \text{div} \vb{F} \dd{A}}\) is the flux across sides of rectangle
Calculating the flux over each side
- Flux across top: \(\approx (\vb{F}(x, y + \Delta y) \cdot \vu{j}) \Delta x = N(x,y+\Delta y) \Delta x\)
- Flux across bottom: \(\approx (\vb{F}(x, y) \cdot - \vu{j}) \Delta x = - N(x,y) \Delta x\)
- Adding up (top and bottom):
- \({\displaystyle \approx (N(x,y + \Delta y) - N(x,y)) \Delta x \approx \biggl(\pdv{N}{y} \Delta y \biggr) \Delta x}\)
- Similarly for left and right
- \({\displaystyle \approx (M(x + \Delta x,y) - M(x,y)) \Delta y \approx \biggl(\pdv{N}{x} \Delta x \biggr) \Delta y}\)
- Adding up all sides
- \({\displaystyle \approx \biggl(\pdv{M}{x} + \pdv{N}{y} \biggr) \Delta x \Delta y}\)
If the total flux over the sides of the small rectangle is positive, this means that there is a net flow out of the rectangle. For example, if there is a source adding fluid directly to the rectangle, as pouring water in the rectangle (According to conservation of matter).
If the total flux is negative, there is a net flow into the rectangle. So this implies there is a sink withdrawing fluid from the rectangle. (Negative source)
The source rate for the rectangle is the flux over sides of rectangle, so we can say that
- \({\displaystyle \text{source rate for the rectangle} = \biggl(\pdv{M}{x} + \pdv{N}{y} \biggr) \Delta A}\)
- \({\displaystyle \text{source rate at }(x,y) = \biggl(\pdv{M}{x} + \pdv{N}{y} \biggr) = \text{div} \vb{F}}\)
- \({\displaystyle \text{source rate for }R = \iint_R \text{div} \vb{F} \dd{A}}\)
8 How can interpret physically Green’s Theorem in the normal form?
8.1 Front
How can interpret physically Green’s Theorem in the normal form?
8.2 Back
\begin{align*} \qq{total flux across} C &= \qq{source rate for} R \\\ \oint_C M \dd{y} - N \dd{x} &= \iint_R \biggl(\pdv{M}{x} + \pdv{N}{y} \biggr) \dd{A} \end{align*}
9 What is the work by \(\vb{F}\) around
9.1 Front
What is the work by $\vb{F}$ around $C$
Let \(\vb{F}\) a two dimensional vector field and any curve
9.2 Back
\({\displaystyle \oint_C \vb{F} \dd{\vb{r}} = \oint_C M \dd{x} + N \dd{y}}\)
10 What is the source rate for
10.1 Front
What is the source rate for $R$
Let be a region closed by a curve , and \(\vb{F}\) a two dimensional vector field
10.2 Back
Source rate for is \({\displaystyle \iint_R \text{div} \vb{F} \dd{A}}\)
Using the Green’s Theorem in the vector form, the flux of \(\vb{F}\) across is the source rate for
\({\displaystyle \oint_C \vb{F} \vu{n} \dd{s} = \int_R \text{div} \vb{F} \dd{A}}\)
11 Compute the flux of \(\vb{F}\) across using geometric methods
11.1 Front
Compute the flux of $\vb{F}$ across $C$ using geometric methods
Let \(\vb{F} = g( r) \ev{x,y}\), where is a function of the radial distance . is the circle of radius centered at the origin and traversed in a clockwise direction
11.2 Back
\(\vb{F}\) is a radial field, and \(\vu{n}\) is parallel in the opposite direction, so we have \(\vb{F} \vu{n} = -g(a) \cdot a\) at the circle.
Flux = \(\int_0^{2 \pi} -g(a)a \dd{s} = -g(a) 2 \pi a^2\)
12 Is this vector field conservative
12.1 Front
Is this vector field conservative
Let \(\vb{F} = r^n \ev{x,y}\) for any integer
12.2 Back
This is radial vector field, as is not defined at the origin when
\(\vb{F} = \ev{M,N}\), of all values of except the origin and , you can apply Green’s Theorem directly.
As , so \(\text{curl} \vb{F} = 0\)
Using extended Green’s Theorem,
- \({\displaystyle \oint_{C_1} \vb{F} \dd{\vb{r}} = 0}\)
- \({\displaystyle \oint_{C_3} \vb{F} \dd{\vb{r}} = 0}\) (it’s a circle)
Any region between and , the \(\iint_R \text{curl} \vb{F} = 0\), so this vector field is conservative
13 Where can we say the \(\vb{F}\) is conservative?
13.1 Front
Where can we say the F is conservative?
Let \({\displaystyle \vb{F} = \frac{-y \vu{i} + x \vu{j}}{x^2 + y^2}}\)
What happens at point ?
13.2 Back
\(\vb{F}\) is not defined at origin, but \(\text{curl} \vb{F} = 0\) everywhere else
Domain is the plane less origin: not simply connected
At can’t use Green’s Theorem directly, so we need to use an expansion of this theorem.
Where in the region inside and , \({\displaystyle \iint_R \text{curl} \vb{F} \dd{A} = 0 = \oint_{C_2} \vb{F} \cdot \dd{\vb{r}} - \oint_{C_3} \vb{F} \cdot \dd{\vb{r}}}\)
This implies that \({\displaystyle \oint_{C_2} \vb{F} \cdot \dd{\vb{r}} = \oint_{C_3} \vb{F} \cdot \dd{\vb{r}}}\).
As is a circle, we can compute this line integral geometrically.
\({\displaystyle \vb{F} \cdot \vu{T} = 1/a \implies \oint_{C_3} \vb{F} \cdot \dd{\vb{r}} = \int_{C_3} \frac{1}{a} \dd{s} = \frac{2 \pi a}{a} = 2 \pi}\)