<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/95228166-d4d7-4561-a3ee-32a77b19611b/fields.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/95228166-d4d7-4561-a3ee-32a77b19611b/fields.png" width="40px" /> Field: a function that assigns a value to every point in space. Scalar fields assign one value to every point, while vector fields assign a vector to every point. Example: The vector field: $V=\frac{1}{\sqrt{x^2+y^2}}\left( -y\hat i + x \hat j \right)$

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Vector lines

• For vector fields we will often draw “field-lines”. These are tangential to the field for the field above it would look like:
• If we have a vector field $V=V_x \hat i+V_y\hat j$ the equation of field lines is:

$$ \frac{\text dy}{\text dx}=\frac{V_y}{V_x} $$

• which can be integrated to get the equation of field-line curves: $y=f(x)+c$ where the different c give the different field lines

Gradient of a scalar field

<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/109be4ce-5f19-4600-829c-b1d40ecb53aa/red-triangle-pointed-down_1f53b.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/109be4ce-5f19-4600-829c-b1d40ecb53aa/red-triangle-pointed-down_1f53b.png" width="40px" /> $\nabla$, gradient (or del) operator: is defined as:

$$ \vec \nabla=\left( \frac{\partial}{\partial x} \hat i + \frac{\partial}{\partial y} \hat j + \frac{\partial}{\partial z} \hat k\right) $$

It is applied to a function as follows:

$$ \vec \nabla \psi =\left( \frac{\partial\psi}{\partial x} \hat i + \frac{\partial\psi}{\partial y} \hat j + \frac{\partial\psi}{\partial z} \hat k\right) $$

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• The change in a field $\text d \psi$ is given by:

$$ \text d\psi = \frac{\partial \psi}{\partial x} \text dx + \frac{\partial \psi}{\partial y} \text dy + \frac{\partial \psi}{\partial z} \text dz= \vec \nabla\psi \cdot\text d \vec{r} $$

• Taking the grad operator of a scalar field gives a vector field

Properties of $\vec \nabla$

The 4 key properties:

• $\vec \nabla f$ is a vector field
• $\vec \nabla f$ defines the maximum rate of change of $f$ (both magnitude and direction)
• The direction of $\vec \nabla f$ is perpendicular to the contours of constant $f$
• The unit vector normal to a level-surface is: $\frac{\vec \nabla f}{|\vec \nabla f|}$

Practical uses:

• Defining the direction of the maximum rate of change of a scalar field
• Calculating the rate of change of a scalar field in a given direction
• Calculating the unit vector $\hat n$ normal to a level surface

Example: of typical questions:

1. Find the unit vector $(x_0,y_0,z_0)$ which is normal to the level surface $f(x,y,z)=$ constant.

   $$ \hat n=\frac{\vec n}{|\vec n|}=\frac{\vec \nabla [f(x_0,y_0,z_0)]}{\left |\vec \nabla [f(x_0,y_0,z_0)]\right |} $$

2. Find the rate of increase of the field $f$ at $(x_0,y_0,z_0)$ in the direction between $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$

   $$ \begin{aligned} \frac{\text df}{\text d s}&=\vec \nabla f\cdot\hat u=\vec \nabla [f(x_0,y_0,z_0)]\cdot\frac{\text d \vec s}{|\text d \vec s|} \\ &=\vec \nabla [f(x_0,y_0,z_0)]\cdot\frac{[x_2-x_1]\,\hat i+[y_2-y_1]\,\hat j+ [z_2-z_1]\,\hat k}{\left|[x_2-x_1]\,\hat i+[y_2-y_1]\,\hat j+ [z_2-z_1]\,\hat k \right|} \end{aligned} $$

$\vec \nabla$ in polar coordinates

• In cylindrical polar coordinates:

$$ \vec{\nabla}\psi = \frac{\partial\psi}{\partial r}\,\hat{r} +\frac{1}{r}\frac{\partial\psi}{\partial \theta}\,\hat{\theta} +\frac{\partial\psi}{\partial z}\,\hat{z} $$