This Downward Pointing Triangle Means Grad Div and Curl in Vector Calculus (Nabla / Del) by Parth G

Gradient, Divergence, and Curl are extremely useful operators in the field of Vector Calculus. In this video, we'll be trying to get an intuitive understanding of what they represent, as visually as possible.

Hey everyone, in this video I wanted to put a physicist's spin on a heavy mathematical topic. Gradient, Divergence, and Curl are used in many different topics in physics. For example, we see them in the study of gravitation, electromagnetism, relativity, and many more.

The del or nabla operator, represented by a downward-pointing triangle, can be thought of as a vector of partial derivatives. In three dimensions, our vector consists of partial d/dx, partial d/dy, and partial d/dz. What each one of these represents is the rate of change of any given quantity, to which we apply del, in the direction x, y, or z. The partial derivative simply represents the fact that we imagine everything in any other direction to be constant. So if we are trying to find partial d/dx, then we assume our quantity does not change in y and z, and so on.

This vector can be applied directly to a scalar field in order to find its so called "gradient", often shortened to "grad". Now a scalar field is basically just any region of space (whether real or abstract) that can be assigned some number or quantity. For example, on a map we may see numbers or contours representing the height above sea level of each point. This is a scalar field, because each point can be given a number - its height above sea level. And when we apply the gradient operator to our scalar field, what we get in return is a vector field. This vector field represents the rate of fastest change of the original scalar field at every point. In other words, for our map analogy, at every point the gradient points up the steepest climb adjacent to it. And each vector's size represents exactly how steep that climb is. (By the way, a vector field is just a region of space in which we can assign a vector to each point - size and direction).

Additionally, we can apply the nabla operator to a vector field, if instead of direct application, we choose to take the dot product between our del and the field. Now a dot product, or a scalar product, between two vectors, simply consists of multiplying corresponding components of each vector and then adding up these products. If our vector field describes the electric field generated in space by nearby charges, then taking the dot product between the del and our field, gives us what's known as the "divergence" of the field. This is often shortened to "div", and tells us exactly how much of the field is being emitted or absorbed by each point in our region of space. In other words, it is a measure of exactly how much each point is a source or a sink of the field. And as it turns out, only points where there are charges, can be sources or sinks. Positive charges are sources of the electric field, the field seems to emanate from them, and negative charges are sinks because the field lines end there. This is determined by one of Maxwell's Equations. So basically, finding the divergence of a vector field results in a scalar field.

Finally, we can also find the cross product between the del operator and a vector field. A cross product, or vector product, usually refers to a measure of alignment between two ordinary vectors. The end result is a third vector, perpendicular to both the originals, and this vector will be as long as possible if the two originals are exactly at right angles to each other. But if they are aligned or anti-aligned, then the resultant vector will have a length of zero. However, taking the cross product between the nabla and a scalar field measures the "circulation" of the original field. At each point, we find a vector that represents how much a little stick / piece of plastic would rotate if placed at that point in the field. If our field represented the flow of water in a lake, then we could imagine putting a stick into it and in some regions it would spin. In these regions, we could represent the "curl" of the water flow field as another vector, pointing along the axis of our stick's rotation. And the more the stick rotates, the larger the vector. If it rotates clockwise, the vector points downward, and if it rotates anticlockwise, the vector points upwards.

Timestamps
0:55 - Nabla / Del and Partial Derivatives
3:21 - Scalar Fields and Gradient
5:08 - Vector Fields and Divergence
8:50 - Curl
11:04 - Applications (in Physics)

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