Earth’s magnetic field A cross section of Earth’s magnetic field can be represented as a vector field in which the center of Earth is located at the origin and the positive y-axis points in the direction of the magnetic north pole. The equation for this field is = where m is the ma...
In vector calculus there is an operator called del which is actually the greek letter nabla (∇). This operator looks like an upside down upper-case delta, but it performs a much different operation. The del operator is what is known as a vector operator. It is similar to a vector, e...
Calculus Function Line integral Mean Potential Terms Vector Vector calculus In summary: V represents is only non-zero when we move from one constant surface to a different one. Line integrals are usually used to determine work. If V is a gravitational field, for example, then these surfaces wo...
Machine learning is a field of artificial intelligence that allows systems to learn and improve from experience without being explicitly programmed. It has become an increasingly popular topic in recent years due to the many practical applications it has in a variety of industries. In this blog, w...
In summary, the work done by the force field F(x, y, z) = 3xi +3yj + 7k on a particle moving along the helix r(t) = 4 cos(t)i + 4 sin(t)j + 4tk, 0 ≤ t ≤ 2π is 56π. This is found by taking the circulation of the vector field along the...
This looks like a rather strange system; but it is three vector equations (3), (4), (5) in four vector (or one-form) unknowns , together with a divergence-free condition (6) and a positive definiteness condition (7), which I view as basically being scalar conditions. Thus, this ...
This is fairly well known, but it took a little effort for me to locate the required arguments, so I am recording the calculations here. The first observation is that every unit quaternion induces a unit tangent vector on the unit sphere , located at ; the third unit vector is then ...
What Is a Gradient? A gradient is a derivative of a function that has more than one input variable. It is a term used to refer to the derivative of a function from the perspective of the field of linear algebra. Specifically when linear algebra meets calculus, called vector calculus. The...
When performing iterative improvement, is a vector whose elements are the difference of nearby inexact floating-point numbers, and so can suffer from catastrophic cancellation. Thus iterative improvement is not very useful unless = Ax(1) - b is computed in double precision. Once again, this is ...
(There are ways to make the use of infinitesimals rigorous, such as non-standard analysis, but this is not the focus of my post today.) In single variable calculus, we learn that if we want to differentiate a function at some point x, then we need to compare the value f(x) of f ...