Ok. a couple of things you need to know first. Do you know that ax, ay, and az are unit vectors in the direction of the x, y, and z axes respectively? Do you know the × operation is the vector cross product? Do you know how to evaluate a vector cross product? When you can answer those three questions we can proceed.
OK. So let us take the first one. A differential length dl is the summation of the length dx times the unit vector in the x direction ax, plus the length dy times the unit vector in the y direction ay, plus the length dz times the unit vector in the z direction. This vector sum produces a vector of the correct length with an orientation that depends on the magnitudes of the scaler quantities dx, dy, and dz. Are we OK with that?
The differential volume element is also pretty easy to grasp since it is just a cube of infinitesimally small size made up of sides with infinitesimally small lengths, namely dx, dy, and dz. Area is the tough one to grasp, until you realize that the cross product of two vectors produces a pseudo vector that is perpendicular to both, and which has a magnitude proportional to the differential area with sides dx by dy, or dx by dz, or dy by dz. Are you still with me? https://en.wikipedia.org/wiki/Pseudovector https://en.wikipedia.org/wiki/Poynting_vector
ds1 has a magnitude of dy dz and a direction of ±ax, where ax is the unit vector in the x direction. Similarly for the other two directions.
Wow. Interesting interplay in this thread. Makes me realize how hard it would be to be a calculus teacher. Once you have it and use it all the time it becomes second nature. It is in trying to explain ideas like curl that I am glad I chose engineering over education. Good job PapaBravo. In some forums people aren't always so patient and willing to stay with it.
To be fair, I didn't get many of the concepts with my first bite at the apple -- so to speak. It wasn't until I took Fluid Dynamics that these concepts became cemented inside my thick skull. I really really loved that class.
Do you have any insight into polarization density, permittivity, and the D-field that might help me to make it click? Getting bogged down in my E&M class.
Nothing comes to mind, except divide an conquer. More pointed questions would be helpful to understand what is giving you trouble.
I never did any projects on my own that I retained the rights to post and publish. Many of them were done in the era from 1969 to 1992 on paper which got left behind each time I changed employers.