00:01
Consider the scalar function f of t and x and a vector function a of t and x.
00:08
We want to determine which of the following expressions are mathematically well defined.
00:13
So here we have a total of five expressions that include gradients, divergencies, curls of our scalar or vector function.
00:27
I want to know what are their results.
00:29
Do they even exist? so first, let me summarize.
00:34
What is the gradient, the divergence, and curl of a function? so the gradient is written as the nabla operator times a scalar function f, where the nabla operator is a vector containing the partial derivatives.
01:09
This is for a 3d case, so we have our partial derivatives in the xyz direction.
01:18
This is a vector here times f.
01:22
And if we were to develop this, we will obtain the partial derivative f with respect to x in the i direction, plus the partial derivative of f with respect to y in the j direction, plus the partial derivative of f with respect to z in the k direction.
01:47
Next, the divergence is written as the dot product of the nabla operator to a vector field, which corresponds to our nabla operator, dot, our scalar field.
02:15
So this number is a scalar that is equal to the partial derivative of our vector field with effect to x plus the derivative of our vector field a with respect to y, plus the derivative of our vector field with f to z.
02:46
And this here is a scalar, as opposed to the gradient, which gives us a vector.
02:51
Now we know that the gradient is only defined if you apply it to a scalar field, but the divergence is only defined if you apply it to a vector field.
03:01
It doesn't make sense to apply a gradient to a scalar field.
03:07
Next, we have the curl that is defined as the nabla operator.
03:13
Cross some vector field a.
03:20
So writing out the terms will be equal to our nabla operator, or the vector here, of our nabo operator, cross our vector field a, which will be equal to the determinant of the vector i, j, k, with our nabo operator of the second row, and our component of our vector field in the third row.
04:03
I'm not going to develop this term, but essentially this returns a vector.
04:10
So a curl gives us a vector, the divergence gives us a scalar, and the gradient returns a vector.
04:26
And something to keep in mind is that the gradient is only defined once applied to a scalar field, the divergence is only defined once applied to a vector field, and similarly the curl is only defined when applied to a vector field.
04:41
So now having, i guess, a summary of how these operators apply.
04:49
Let's look at our first expression.
04:52
So here we have the divergence applied to a gradient.
04:58
And here our gradient that's applied to a scalar field.
05:01
So this is defined.
05:02
So this here is a vector, a vector field, which is defined when it is a vector field.
05:11
Applied with a divergence.
05:14
So this here is well defined.
05:17
In fact, this is commonly written as the laplacian of a scalar field.
05:31
The placian is commonly used in differential equations.
05:36
So this expression a is well defined.
05:39
Our second expression, we have the curl applied to a gradient of a scalar field...