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Vectors are directional numbers. Calculus determines changes. Electromagnetism involves directional fields and thus vector calculus is the tool to calculate the changes in directional fields.
The training in Mathematics and Physics is deficient in that Nature involves the combination of real and vector numbers called Quaternions. Quaternions were invented by William Rowan Hamilton in 1843. Quaternions consist of a real number r and three vectors (i,j,k) such that i2 = j2 = k2 = ijk = -1.
A quaternion point is p=r + ix +jy + kz= r + v where v is the vector part.
Quaternion calculus has a derivative I call X for Khepra which consists of Hamilton's vector derivative called Del = id/dx + j d/dy + kd/dz and a real derivative d/dr = d/cdt .
X= d/dr + Del = d/dr + id/dx + jd/dy + kd/dz = d/cdt + Del = [d/dr,Del]
Using this quaternion derivative the fundamental laws of electromagnetism can be derived as th Boundary Condition, 0= XE where E is the quaternion electric field E=Er + Ev = [Er,Ev].
The First Derivative of the Electric field is
XE= (dEr/cdt - Del.Ev) + (dEv/cdt + DelxEv + Del Er)
The Equilibrium Condition for the Electric field occurs when the the First Derivative is set to zero:
0=XE= (dBr/dt - Del.Ev) + (dBv/dt + Del Er)
This is the Quaternion Equilibrium Condition Equation, notice that the Curl Term DelxEv =0 and is not in the equation. The curl is zero at Equilibrium and the remaining vector terms are "Equal and Opposite"! Equilibrium requires that the sum of the reals and vectors sum to zero. The vectors cannot sum to zero unless DelxEv=0, this happens only when the other terms are parallel or anti-parallel. Equilibrium is the anti-parallel case, thus Newton's "Equal and Opposite" Rule in his 3rd law of Motion.
This Equilibrium Condition is the Stationary and Invariant Condition and the Cauchy-Riemann Continutiy Condition.
Maxwell's Equations
dBr/dt - Del.Ev=0
dBv/dt + DelxEv=0
are incorrect in including DelxEv, it should be Del Er. DelxEv is perpendicular to dBv/dt =dEv/cdt. Vector Calculus shows DelxEv is perpendicular to dEv/dr, thus the sum of orthogonal vectors is not zero unless both vectors are zero.
This shows that Maxwell's Equations are incorrect and the proper Electromagnetism Equations are derived by Quaternion Calculus.
What else can I help you with?
What are the engineering applications of vector calculus?
in which field vector calculus is applied deeply
What is application of vector calculus in engineering?
It is used to position an object in3D
How is calculus used in physics?
Physics is an especially heavy user of the calculus: mechanics, electromagnetism, acoustics, optics. Richard Feynman was known for his great facility in this area of mathematics.
How many calculus courses are there?
High SchoolCalculus AB - Calculus 1Calculus BC - Calculus 1 + part of Calculus 2College:Calculus 1: Single variable calculusCalculus 2: Multi-variable CalculusCalculus 3: Vector CalculusCalculus 4: Differential Equation
Is Elementary Calculus the same as Pre-Calculus?
In short, no. Elementary calculus includes finding limits, basic differentiation and integration, dealing with sequences and series, and simple vector operations, among other concepts. Pre-calculus mostly focuses on the algebra necessary to perform those operations, with perhaps some introduction to limits or other simple ideas from elementary calculus.
What are the engineering applications of vector calculus?
in which field vector calculus is applied deeply
What is the differentence between calculus and vector calculus?
Hence the reason for why it is called Vector Calculus, Vector Calc. is simply an expansion in the calculus subject are in math. It deals with Taylor's Formula (in calc 2 you learn the taylor polynomial and the taylor series), theorems from Green, Gauss, and Stokes, and much more.
What has the author Richard H Crowell written?
Richard H. Crowell has written: 'Calculus of vector functions' -- subject(s): Vector analysis 'Calculus with analytic geometry' -- subject(s): Analytic Geometry, Calculus
What has the author H K Nickerson written?
H. K. Nickerson has written: 'Advanced calculus, by H.K. Nickerson, D.C. Spencer and N.E. Steenrod' -- subject(s): Calculus, Vector analysis 'Advanced calculus' -- subject(s): Calculus, Vector analysis
Applications of vector calculus in engineering?
Electromagnetic fields, gravitational fields and fluid flow. If you are an engineer you will come across vector calculus to handle three dimensional space.
Applications of vector calculus in electrical engineering?
Vector calculus is applied in electrical engineering especially with the use of electromagnetics. It is also applied in fluid dynamics, as well as statics.
What is the application for vector calculus?
Measures of motion (displacement, velocity, acceleration) and forces are all vectors so any study involving these would require vector calculus.
What are the electronics and communication engineering applications of vector calculus?
The theory of radio waves and waveguides is explained in terms of equations in the form of vector calculus. Examples are Maxwell's equations.
What is the significance of the area vector in the context of vector calculus and its applications?
The area vector in vector calculus represents the direction and magnitude of a surface area. It is important in applications such as calculating flux, which measures the flow of a vector field through a surface. The area vector helps determine the orientation of the surface and is crucial for understanding the behavior of vector fields in three-dimensional space.
What is the significance of the divergence of a scalar times a vector in the context of vector calculus?
The significance of the divergence of a scalar times a vector in vector calculus is that it simplifies to the scalar multiplied by the divergence of the vector. This property is important in understanding how scalar fields interact with vector fields and helps in analyzing the flow and behavior of physical quantities in various fields of science and engineering.
Why vector calculus are used in mechanical engineering?
Mechanical engineering usually deals with forces and their effects on materials. Forces are vectors and so, to study their effects you need to use vector calculus.
What is the relationship between the gradient dot product and vector calculus?
The gradient dot product is a key concept in vector calculus. It involves taking the dot product of the gradient operator with a vector field. This operation helps in understanding the rate of change of a scalar field in a given direction. In vector calculus, the gradient dot product is used to calculate directional derivatives and study the behavior of vector fields in three-dimensional space.
