If you have come across electromagnetics then you shold be aware of Curl
Curl expression can be got by evalutating the determinent
In this think the 2 edges connecting floor and wall as z and x (the edge connecting floor and the wall that having window is z and other is y) and the edge connecting two walls as y axis. so now floor is z-x plane and the wall having window is z-y plane and the other wall adjacent to wall with window is x-y plane. Now we will visualize what this vx, vy and vz all about. vx is a field in the direction of x axis. Field can be anything but for getting insight we will consider that as an air flow (speed of the air). If air is flowing from window then the direction of this air flow is in x direction because it will in the same direction as x axis.
Now for our getting insight about curl we will assume Vy = Vz = 0 so only Vx > 0 and Vx is the speed of air flow which increases as we move in Z direction so we will get
dVx/dz in y direction - dVx/dy in z direction.
If we assume that change in Vx when we are moving along y is constant. SO dVx/dy is zero and we are left with
dVx/dz in y direction
If Vx increases if you walk along positive z direction (i.e if you consider Vx as speed of air flow and if the air speed increase if you are moving along z direction) dVx/dz will be positive.
Are you wondering what meaning this curl's + ve value (increasing value of Vx) of dVx/dz is having. Take a pen or scale or rod on a table place your right index finger in the right end of the rod/pen and place your left hand index finger on other corner of the pen. Now move left side of the pen with low speed and force and move your right end of the pen with high speed. Did you noticed the pen is actually trying to rotate? this is what the curl will give.
Curl expression can be got by evalutating the determinent
The first row of this determinant is the direction of the resultant vector. so if we evaluiate this deterterminant we will get.
(dvz/dy - dvy-dz) in x direction - (dvz/dx - dvx/dz) in y direction + (dvy/dx - dvx/dy) in z direction
So what this expression says we will look this in detail. Before that we will try to visualize the component in the determinant. so we will start with axis. visualize corner of teh room like the one shown below.
(dvz/dy - dvy-dz) in x direction - (dvz/dx - dvx/dz) in y direction + (dvy/dx - dvx/dy) in z direction
So what this expression says we will look this in detail. Before that we will try to visualize the component in the determinant. so we will start with axis. visualize corner of teh room like the one shown below.
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In this think the 2 edges connecting floor and wall as z and x (the edge connecting floor and the wall that having window is z and other is y) and the edge connecting two walls as y axis. so now floor is z-x plane and the wall having window is z-y plane and the other wall adjacent to wall with window is x-y plane. Now we will visualize what this vx, vy and vz all about. vx is a field in the direction of x axis. Field can be anything but for getting insight we will consider that as an air flow (speed of the air). If air is flowing from window then the direction of this air flow is in x direction because it will in the same direction as x axis.
Now for our getting insight about curl we will assume Vy = Vz = 0 so only Vx > 0 and Vx is the speed of air flow which increases as we move in Z direction so we will get
dVx/dz in y direction - dVx/dy in z direction.
If we assume that change in Vx when we are moving along y is constant. SO dVx/dy is zero and we are left with
dVx/dz in y direction
If Vx increases if you walk along positive z direction (i.e if you consider Vx as speed of air flow and if the air speed increase if you are moving along z direction) dVx/dz will be positive.
Are you wondering what meaning this curl's + ve value (increasing value of Vx) of dVx/dz is having. Take a pen or scale or rod on a table place your right index finger in the right end of the rod/pen and place your left hand index finger on other corner of the pen. Now move left side of the pen with low speed and force and move your right end of the pen with high speed. Did you noticed the pen is actually trying to rotate? this is what the curl will give.
dVx/dz in y direction
what about the term "in y direction" If Vx increases when you move along +ve z direction Vx will rotate and if you notice the axis of rotation will be Y axis.
what about the term "in y direction" If Vx increases when you move along +ve z direction Vx will rotate and if you notice the axis of rotation will be Y axis.
If you input the field vector to the curl it will tell you whether the given field will rotate? if so in which direction?
