So this Pencast will review vector calculus.
And so the first thing one introduces in vector calculus
is the concept of a differential operator, which is called del.
And so it's symbolized by this triangle here.
And what this is-- formally, it can be written as x hat d by dx,
plus y hat d by dy plus z hat d by dz, in Cartesian coordinates.
And so, by operator, what we mean is it's something
that doesn't have any meaning in and of itself,
but it can act on different functions.
And so, if we let it act on a scalar, for instance, f of x, y,
z, what we get back is a vector field, which is referred to as the gradient.
And so what the vector field ends up being is
it's going to be partial df by dx, x hat, plus partial df by dy,
y hat, plus partial df by dz, z hat.
And so, as an example, if one takes f of xyz
to be equal to x squared plus y squared plus z squared minus some constant,
r squared, this is the equation of a hemisphere
if we restrict to the upper half plane.
Or actually, probably one word.
And so it's drawn here.
And so this is the set of all points whose coordinate-- x squared,
y squared, z squared-- is equal to r squared.
And so if we take the gradient of f for the sphere--
and you should verify for yourselves the operation-- what you get
is 2x times x hat plus 2y y hat plus 2z z hat.
And so the point is, this function here is
a scalar, which describes the sphere.
And what you get back here is a vector or a vector field.
And so this vector field assigns to each point on the sphere, a vector.
And the gradient, what's important to know about it is it
points uphill for the function.
So essentially, it points in the direction
in which the function is increasing.
So in this case, it generates a vector field-- and I'll draw it
by these little arrows, which are pointed up along the sphere
or along the hemisphere.
And so that's the first operation, which is the gradient.
Now, we can do other things with this operator, del.
And so the next operation is called the divergence.
And so this occurs if we take the operator del
and we let it act on some vector field-- let's call it a of xyz.
And now, it takes a vector and returns a scalar,
which is given by da by dx, plus da by dy plus da by dz.
And so, we can define another operation, which is called the curl.
And so what this is is it takes a vector and returns another vector.
And so the curl is defined by taking this operator, del,
and forming the cross-product.
And so what you'll get here-- sorry, we should have a subscript, a sub x,
a sub y, a sub z.
And so, what this will do is you simply take the del operator and form
the cross-product.
So what you'll get here is the x component
is going to be da sub z, partial with respect to y, minus da sub y,
with a respect to z.
And that's the x hat component.
And then you have minus da sub z, with respect
to x, minus dax, with respect to z.
And that's the y hat component.
And then you have plus da sub x, with respect to y-- sorry,
da sub y, with respect to x, minus da sub x, with respect to y.
And that's the z hat component.
So to get some intuitive feel for what these things are,
rather than just mathematical expressions--
if we take a vector field, which is the electric field of a charge.
Which you might imagine from physics.
So this would be given by minus q times k
over r squared times the unit vector pointing in the radial direction.
And we'll talk about polar coordinates in the next Pencast.
But so, the vector field looks like this.
And so, essentially what this is is you see that you
have lines all diverging from a point.
And so this is an example of a vector field, e, where the divergence of e
is not equal to 0.
But if you were to take the curl of e, in this case, you would find 0.
So this is a vector field whose divergence is not 0,
but its curl is 0 everywhere.
And so intuitively, it means that the lines of the vector field
are diverging from a point everywhere, but there's no rotation to them.
In contrast, a magnetic field, for instance, which is given by,
for instance, mu 0 i, over 2 pi r.
And you might remember from physics that this is a vector field.
The magnetic field generated by a wire, which would be coming out of the page
by the right-hand rule.
So if we put a wire here at the origin-- and this theta hat
is a unit vector which points in the angular direction.
And so this is a vector field whose divergence is 0 everywhere,
but the curl is not 0.
So this is a vector field-- is not equal to 0.
So this is a vector field that has a curl but no divergence.
So intuitively, what the curl measures is the amount
of rotation in the vector field.
So if you imagine a vortex, like a hurricane or a tornado,
that's something that has a curl but no divergence.
Whereas something like a drain-- a source or a sink--
is something that has a divergence but no curl.
And so if you don't totally remember these operations,
it might be good to go back and review a calculus book.
And we'll use these periodically throughout the course,
particularly the gradient more than anything else.