Field and Gradient

Transcribed Lecture Notes (First 3 Pages)

Fields of Force: The Physical Power of Gradient

Field: A physical quantity that assigns a value to every single point in space and time.

Scalar Field: The simplest way nature populates space by assigning a single, solitary number to every point in the space.

Imagine the room you are sitting in right now. Pick any random point in space, say a few inches away from your nose. That specific point has a definite Temperature (TT). If you move your hand closer to a window, Temperature changes.

If you mapped the temperature at every single point / co-ordinate point (x,y,z)(x, y, z) in the room, we can write this function:

T(x,y,z)T(x, y, z)

Vector Field: A rushing river or swirling cyclone. If you pick any point, it has a specific velocity as well.

V(x,y,z)=Vx(x,y,z) e^x+Vy(x,y,z) e^y+Vz(x,y,z) e^z\mathbf{V}(x, y, z) = V_x(x, y, z)\,\hat{e}_x + V_y(x, y, z)\,\hat{e}_y + V_z(x, y, z)\,\hat{e}_z

Gradient

Imagine you are standing inside a 3D scalar field, such a room with complex temperature distribution T(x,y,z)T(x, y, z).

You decide to take an infinitesimal step in arbitrary direction. We represent your tiny step as a displacement vector:

dr⃗=dx e^x+dy e^y+dz e^zd\vec{r} = dx\,\hat{e}_x + dy\,\hat{e}_y + dz\,\hat{e}_z

Change in temperature due to moving tiny step is also tiny:

dT=∂T∂x dx+∂T∂y dy+∂T∂z dzdT = \frac{\partial T}{\partial x}\,dx + \frac{\partial T}{\partial y}\,dy + \frac{\partial T}{\partial z}\,dz
=(∂T∂x e^x+∂T∂y e^y+∂T∂z e^z)⋅(dx e^x+dy e^y+dz e^z)= \left(\frac{\partial T}{\partial x}\,\hat{e}_x + \frac{\partial T}{\partial y}\,\hat{e}_y + \frac{\partial T}{\partial z}\,\hat{e}_z\right) \cdot \left(dx\,\hat{e}_x + dy\,\hat{e}_y + dz\,\hat{e}_z\right)
=(∂∂x e^x+∂∂y e^y+∂∂z e^z)⏟↓Gradient / del operator / nabla / ∇T⋅(dx e^x+dy e^y+dz e^z)= \underbrace{\left(\frac{\partial}{\partial x}\,\hat{e}_x + \frac{\partial}{\partial y}\,\hat{e}_y + \frac{\partial}{\partial z}\,\hat{e}_z\right)}_{\substack{\downarrow \\ \text{Gradient / del operator / nabla / } \nabla}} T \cdot \left(dx\,\hat{e}_x + dy\,\hat{e}_y + dz\,\hat{e}_z\right)
=∇T⋅(dr⃗)= \nabla T \cdot (d\vec{r})
Download Original PDF