Scalar and Vector Fields

Transcript

[Music] in advanced physics we frequently talk about fields field is a fancy word for function in three dimensions the idea of a field is that the function has a value at each point in space at the same time so it's different from a trajectory like you've studied before because in a trajectory a particle is visiting each point along a path in space in a field we assign a value to every point in three-dimensional space there is no concept of a path because the field is everywhere for example the temperature in a room or the brightness from a light bulb or the electric potential of an electric dipole are all scalar fields we can visualize the value of the field at each point using color since we've already used all three spatial dimensions to represent the space of the problem to be studied when we work with scalar fields we often need to apply a three-dimensional version of the derivative called the gradient the gradient of a scalar field gives the three-dimensional slope of the field at each point in space the gradient tells you what direction is uphill at each point you can also say that the gradient always points from the region of lower value to the region of higher value for example with a temperature field the gradient will tell you where the heat sources are with the electric dipole the gradient goes around from one charge to the other [Music] but sometimes you want to know the derivative in a particular direction for this we need our friend from the previous video the dot product if you take the dot product of a vector with the gradient at a point in space you can find the slope of the scalar field at that point in that direction the directional derivative is an answer to the question of if i move in this direction what will the slope be finally there's also a second derivative in three dimensions called the laplacian you simply take two derivatives in each cartesian direction the laplacian of a scalar field tells you the 3d concavity of the scalar field at each point for example for the electric dipole the laplacian shows you where the charges are located and which regions contain empty space a vector field is also a function where the input is a point in space but the output is a vector a prime example of a vector field is water flowing around in a pond each water molecule has its own velocity vector which creates a vector field over the whole pond or there's the electric field of a dipole the two charges emit electric fields everywhere in space at each point in space the electric field can have a different magnitude and direction or different x y and z components or there's the magnetic field around two wires carrying current these vectors form a pattern that swirls around the two currents and this pattern will change if we alter the current in these wires one of the properties of a vector field that we want to study is the divergence the divergence is a type of derivative that tells you where the vectors are spreading out or collapsing together in our water flow example divergence would tell you where the water is being poured in or being poured out for the electric dipole the divergence tells you where the charges are located divergence is calculated as the dot product between the gradient and the vector field we also need to study the curl of a vector field which is a type of derivative that tells us where the vectors are swirling in our water flow example the curl would tell you where the water is forming whirlpools for the pair of wires the curl tells you where the currents are located curl is calculated as the cross product between the gradient and the vector field finally we can also take the laplacian of a vector field which tells you the curvature of the field all three of these types of derivative divergence curl and laplacian show up in different physics applications