3.3 Divergence, gradient, and curl

Divergence, gradient and curl

Assume we have measured the temperature in a room along an axis $x$ . If we wanted to find the temperature change as we move to postion $(x+\Delta x)$ then from the fundamental definition of a derivative we know that is: $$\Delta T=\frac{dT}{dx}\Delta x$$

We can easily extend this concept to 3 dimensions At position $(x,y,z)$ there is a temperature $T(x,y,z)$ . Suppose we then want to find the temperature at $\vec{R}+\Delta \vec{R}=(x+\Delta x,y+\Delta y,z+\Delta z)$ . Then we canuse: $$\Delta T=\frac{\partial T}{\partial x}\Delta x+\frac{\partial T}{\partial y}\Delta y+\frac{\partial T}{\partial z}\Delta z$$

We could define a vector $$\left(\frac{\partial T}{\partial x},\frac{\partial T}{\partial y},\frac{\partial T}{\partial z}\right)$$ and then say $$\Delta T=\left(\frac{\partial T}{\partial x},\frac{\partial T}{\partial y},\frac{\partial T}{\partial z}\right)\cdot \Delta \vec{R}$$ so let's define an operator $$\vec{\nabla }=\left(\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z}\right)$$ Then we can write $$\Delta T=\vec{\nabla }T\cdot \Delta \vec{R}$$ $\vec{\nabla }$ is a vector operator that can be used in other situations involving scalars and vectors. It is often named "del" or "nabla". Operating on a scalar fieldwith this operator is called taking the "gradient" of the field.

We could also operate on a vector field with del. There are two different ways to do this, by taking the dot and the cross products. To operate on a vectorfield by taking its dot product with del is referred to as taking the divergence. ie. $$f=\vec{\nabla }\cdot \vec{h}$$ where $\vec{h}$ is some vector field and $f$ is the resulting scalar field.

Similarly one could take the cross product: $$\vec{g}=\vec{\nabla }\times \vec{h}$$ where $\vec{g}$ is the resulting vector field. $${\displaystyle \begin{array}{c}{g}_x={\left(\vec{\nabla }\times \vec{h}\right)}_x=\frac{\partial h_z}{\partial y}-\frac{\partial h_y}{\partial z}\\ g_y={\left(\vec{\nabla }\times \vec{h}\right)}_y=\frac{\partial h_x}{\partial z}-\frac{\partial h_z}{\partial x}\\ g_z={\left(\vec{\nabla }\times \vec{h}\right)}_z=\frac{\partial h_y}{\partial x}-\frac{\partial h_x}{\partial y}\end{array}}$$ This is referred to as taking the curl of a field.

These operations, Gradient, Divergence and Curl are of fundamental importance. They have been presented above as operations using some newly defined operatorbut they in fact have deep physical significance. When using these operators to express Maxwell's equations in differential form, the meaning of theseoperations will hopefully become more clear. Gradient is the easiest to understand, it can be thought of as a three dimensional slope.

Having defined these operations we can go on to second derivative type things $$\vec{\nabla }\cdot (\vec{\nabla }T)={\nabla }^{2}T=ascalarfield$$ Note that ${\nabla }^2$ occurs so often that is has its own name, Laplacian $$\vec{\nabla }\times (\vec{\nabla }T)=0$$ $$\vec{\nabla }(\vec{\nabla }\cdot \vec{h})=avectorfield$$ $$\vec{\nabla }\cdot (\vec{\nabla }\times \vec{h})=0$$ $$\vec{\nabla }\times (\vec{\nabla }\times \vec{h})=\vec{\nabla }(\vec{\nabla }\cdot \vec{h})-{\nabla }^2\vec{h}$$ $$\vec{\nabla }\cdot \vec{\nabla }\vec{h}={\nabla }^2\vec{h}=avectorfield$$ Here is an example of taking a divergence that will be extremely useful. If $$\vec{r}=x\widehat{ı}+y\widehat{}+z\widehat{k}$$ and $$\vec{k}=k_x\widehat{ı}+k_y\widehat{}+k_z\widehat{k}$$ and

$$\vec{E}=E_x\widehat{ı}+E_y\widehat{}+E_z\widehat{k}$$ then lets find (for $\vec{E}$ is a constant vector) $${\displaystyle \begin{array}{c}\vec{\nabla }\cdot \vec{E}{e}^{i\vec{k}\cdot \vec{r}}=\frac{\partial }{\partial x}{E}_x{e}^{i\vec{k}\cdot \vec{r}}+\frac{\partial }{\partial y}{E}_y{e}^{i\vec{k}\cdot \vec{r}}+\frac{\partial }{\partial z}{E}_z{e}^{i\vec{k}\cdot \vec{r}}\\ =E_x{e}^{i\vec{k}\cdot \vec{r}}\frac{\partial }{\partial x}(i\vec{k}\cdot \vec{r})+\dots \\ =i{E}_x{e}^{i\vec{k}\cdot \vec{r}}\frac{\partial }{\partial x}(k_{x}x+k_{y}y+k_{z}z)+\dots \\ =i{k}_x{E}_x{e}^{i\vec{k}\cdot \vec{r}}+i{k}_y{E}_y{e}^{i\vec{k}\cdot \vec{r}}+i{k}_z{E}_z{e}^{i\vec{k}\cdot \vec{r}}\\ =i\vec{k}\cdot \vec{E}{e}^{i\vec{k}\cdot \vec{r}}\end{array}}$$