A Short Course in Vector Calculus
Want a short course in vector calculus? You can find one in Intermediate Physics for Medicine and Biology.
Divergence
The divergence is defined in Section 4.1 in IPMB, when discussing the continuity equation. The divergence is one way to differentiate a vector field. I our case, the vector field is the current density (or some other type of flux density), j. Its divergence is defined as
When you take the divergence of a vector (a quantity that has both magnitude and direction), you get a scalar (a quantity that has magnitude but no direction). In electrostatics, the electrical charge is conserved, implying that the divergence of the electrical current density is zero.
Curl
The curl is defined in Section 8.6, when analyzing electromagnetic induction. It is another way to differentiate a vector,
The symbols x̂, ŷ, and ẑ are unit vectors, and the vertical lines indicate that you follow the rules for determinants when expanding this expression. The curl appears often when analyzing the magnetic field. In our case, the curl of the electric field equations the negative of the time derivative of the magnetic field (Faraday’s law of induction).
Gradient
The gradient is a way to differentiate a scalar field to get a vector.
You can think of the gradient, ∇, as representing the vector x̂ ∂/∂x + ŷ ∂/∂y + ẑ ∂/∂z. The divergence is then found by taking the dot product of the gradient with a vector, and the curl is found by taking the cross product of the gradient with the vector. In electrostatics, V represents of the electric potential (a scalar) and E represents the electric field (a vector). The two are related by
Laplacian
The Laplacian, ∇^2, is just the dot product of the gradient operator with itself. In other words
You can apply the Laplacian to a vector, but it is more commonly applied to a scalar (such as electrical potential, temperature, or concentration). The Europeans use ∆ to represent the Laplacian, but that’s just weird and we Americans no better than that.
Other Coordinate Systems We have written the divergence, curl, gradient, and Laplacian in Cartesian coordinates. These operators are more complicated in other coordinate systems. Appendix L of IPMB provides expressions for these operators in cylindrical coordinats of spherical coordinates.
The Divergence Theorem The divergence theorem says that the volume integral of div J is equal to the surface integral of the normal component of J. We don’t dwell on this theorem in IPMB, but we do ask the reader to derive it in Homework Problem 4 of Chapter 4.
Stokes’ Theorem We don’t discuss Stokes’ Theorem in IPMB, but I’ve pointed out how we might include a homework problem about it in a previous blog post. Stokes’ theorem says that the line integral of a vector around a closed loop is equal to the surface integral of the curl of that vector of an area bounded by the loop.
So, almost all the big concepts of vector calculus are presented in IPMB. If, however, you want a little more detail, Russ and I recommend the wonderful book div, grad, curl, and all that, by Harry Schey. I learned vector calculus from the first edition of that book as an undergraduate physics major at the University of Kansas. Schey died five years ago, but his book lives on.
Source: http://hobbieroth.blogspot.com/2023/07/a-short-course-in-vector-calculus.html
Anyone can join.
Anyone can contribute.
Anyone can become informed about their world.
"United We Stand" Click Here To Create Your Personal Citizen Journalist Account Today, Be Sure To Invite Your Friends.
Please Help Support BeforeitsNews by trying our Natural Health Products below!
Order by Phone at 888-809-8385 or online at https://mitocopper.com M - F 9am to 5pm EST
Order by Phone at 866-388-7003 or online at https://www.herbanomic.com M - F 9am to 5pm EST
Order by Phone at 866-388-7003 or online at https://www.herbanomics.com M - F 9am to 5pm EST
Humic & Fulvic Trace Minerals Complex - Nature's most important supplement! Vivid Dreams again!
HNEX HydroNano EXtracellular Water - Improve immune system health and reduce inflammation.
Ultimate Clinical Potency Curcumin - Natural pain relief, reduce inflammation and so much more.
MitoCopper - Bioavailable Copper destroys pathogens and gives you more energy. (See Blood Video)
Oxy Powder - Natural Colon Cleanser! Cleans out toxic buildup with oxygen!
Nascent Iodine - Promotes detoxification, mental focus and thyroid health.
Smart Meter Cover - Reduces Smart Meter radiation by 96%! (See Video).