What is the flux of the water through the net?

Now if we rotate the net through an angle $\theta,$ what is the flux through the net?

Now if we rotate the net through an angle $\theta,$ what is the flux through the net?

Before we look at the flux of a vector field through more general surfaces, let's look at flux in two dimensions, by considering the flow of a two dimensional fluid through a curve in the $xy$-plane. Note that in this context of a fluid in $2$D, flux has dimensions area per unit time.

To start, consider the problem of calculating the flux of a fluid with constant velocity $\v = 2~\i$ through a line segment $C$ perpendicular to the flow, where $C$ is given by \[ C = \left\{(x,y)~|~ x = 2, 2\leq y \leq 6\right\}. \]

Constant velocity $\v = 2~\i$ through a line segment $C$ perpendicular to the flow, where $C$ is given by $C = \left\{(x,y)~|~ x = 2, 2\leq y \leq 6\right\}.$

Now consider calculating the flux of the velocity vector $\v(x,y)$ in the $x$-$y$ plane through a curve $C$. We first divide $C$ up into arcs of length $\Delta S$, and approximate $\v$ as constant over each arc.

This constant vector over each arc shall be evaluated at a representative point in each arc, say $P^* = (x^*, y^*)$.

We also approximate the arc as a straight line, so that

The component of $\v$ which is perpendicular to $C$ (over $\Delta S$) is $\approx \v(P^*)\pd \n (P^*)$.

If we take the limit as $\Delta S$, we obtain an exact expression for the flux over the entire curve C as a line integral: \[ \text{Flux}= \int_C \v\pd \n ~dS \] where $\n$ is a unit vector normal to $C$.

If we take the limit as $\Delta S$, we obtain an exact expression for the flux over the entire curve C as a line integral: \[ \text{Flux}= \int_C \v\pd \n ~dS \] where $\n$ is a unit vector normal to $C$.

We use this expression as a definition of flux of any two dimensional vector field $\v$ across a plane curve $C$. Note then that \[ \text{dimensions of flux (in 2D)} = (\text{dimensions of }\v) \times (\text{distance}). \]

To evaluate the $2$D flux integral \[ \int_C \v\pd \n ~dS, \] we need to parameterise $C$ and express all quantities in the line integral in terms of this parameterisation (similar to the computation of line integral in the definition of work done).

1. Parameterise $C$ by finding a $\r(t) = x(t)~\i + y(t)~\j$ with $t \in[a, b]$ that describes $C$.
2. Write $\v(x,y)$ restricted to $C$ as $\v(\r(t))= \v(x(t),y(t))$.
3. Compute a unit tangent vector to $C$ by using $\T(x,y) = \dfrac{\r'(t)}{\norm{\r'(t)}}$, where $\r'(t) = x'(t)~\i + y'(t)~\j$ is a tangent vector to $C$.

4. Be careful of the direction of $\n$. By the definition of vector cross product, and since $\k$ is a unit vector normal to the $x$-$y$ plane, we can take $\n = \T \times \k$.
5. Write $dS = \norm{\r'(t)}dt$.
6. Evaluate the $2$D flux integral as a definite integral in terms of the parameter $t$:
   $\displaystyle \int_C \v \pd \n ~dS= \int_a^b \left[ \v\left(\r(t)\right)\pd \dfrac{\r'(t)\times \k}{\norm{\r'(t)}}\right]\norm{\r'(t)}~dt$
   $\displaystyle \quad\;\,= \int_a^b \v\left(\r(t)\right)\pd \left( \r'(t) \times \k\right) dt$.

1. Parameterise $C$ by finding a $\r(t) = x(t)~\i + y(t)~\j$ with $t \in[a, b]$ that describes $C$.
2. Write $\v(x,y)$ restricted to $C$ as $\v(\r(t))= \v(x(t),y(t))$.
3. Compute a unit tangent vector to $C$ by using $\T(x,y) = \dfrac{\r'(t)}{\norm{\r'(t)}}$, where $\r'(t) = x'(t)~\i + y'(t)~\j$ is a tangent vector to $C$.
4. Be careful of the direction of $\n$. By the definition of vector cross product, and since $\k$ is a unit vector normal to the $x$-$y$ plane, we can take $\n = \T \times \k$.
5. Write $dS = \norm{\r'(t)}dt$.
6. Evaluate the $2$D flux integral as a definite integral in terms of the parameter $t$:
   $\displaystyle \int_C \v \pd \n ~dS= \int_a^b \left[ \v\left(\r(t)\right)\pd \dfrac{\r'(t)\times \k}{\norm{\r'(t)}}\right]\norm{\r'(t)}~dt$
   $\displaystyle \quad\;\,= \int_a^b \v\left(\r(t)\right)\pd \left( \r'(t) \times \k\right) dt$.

Let $C$ be a piecewise-smooth, simple closed curve. Let $v_1(x, y)$, $v_2(x, y)$ be continuous in the region bounded by $C$. (Note that these are some of the conditions of Green's theorem!)

The net outward flux of $\v = v_1~\i + v_2~\j$ across $C$ is given by \[ \text{Net outward flux} = \oint_C \v \pd \n ~dS, \] where $\n$ is a unit vector normal to $C$, directed outward from the region bounded by $C$.