Vectors and Coordinate Systems
... An orthogonal system is one in which the co-ordinates are mutually perpendicular. Nonorthogonal co-ordinate systems are also possible, but their usage is very limited in practice. Let u = constant, v = constant and w = constant represent surfaces in a coordinate system, the surfaces may be curved su ...
... An orthogonal system is one in which the co-ordinates are mutually perpendicular. Nonorthogonal co-ordinate systems are also possible, but their usage is very limited in practice. Let u = constant, v = constant and w = constant represent surfaces in a coordinate system, the surfaces may be curved su ...
Chapter 6 Vocabulary
... Heron’s Area Formula Given any triangle with sides of lengths a, b, and c, the area of the triangle is given by Area = √[s(s-a)(s-b)(s-c)] Where s = (a + b + c) / 2 ...
... Heron’s Area Formula Given any triangle with sides of lengths a, b, and c, the area of the triangle is given by Area = √[s(s-a)(s-b)(s-c)] Where s = (a + b + c) / 2 ...
Gradient, Divergence and Curl: the Basics
... (A) dS = [dAx dx + dAy dy + dAz dz] = closed loop Adr . If the integral around a closed loop is not zero, then that implies that there is some circulation of the vector field. Note that if the curl of the vector is zero everywhere, then there cannot be any circulation of the vector field ...
... (A) dS = [dAx dx + dAy dy + dAz dz] = closed loop Adr . If the integral around a closed loop is not zero, then that implies that there is some circulation of the vector field. Note that if the curl of the vector is zero everywhere, then there cannot be any circulation of the vector field ...
Vectors and Vector Operations
... There are two common ways that one encounters subspaces. The first is the set of all linear combinations of a fixed set of vectors. Proposition 1. Let u1, …, un be a fixed set of vectors and let S consist of all linear combinations of u1, …, un, i.e. i.e. all vectors v such that v = c1u1 + + cnun ...
... There are two common ways that one encounters subspaces. The first is the set of all linear combinations of a fixed set of vectors. Proposition 1. Let u1, …, un be a fixed set of vectors and let S consist of all linear combinations of u1, …, un, i.e. i.e. all vectors v such that v = c1u1 + + cnun ...
Introduction Last year we studied the electric and the magnetic field
... “:” (colon) a : b = c , a divided by b is equal to equals c, the ratio/quotient of a and b is c a/b ( a slash b) , (this fraction can be said as) , a divided by b , a over b Note that “per” is similar in meaning to “divided by” and that “per” is only used for a quantity of “one”, so you can say “per ...
... “:” (colon) a : b = c , a divided by b is equal to equals c, the ratio/quotient of a and b is c a/b ( a slash b) , (this fraction can be said as) , a divided by b , a over b Note that “per” is similar in meaning to “divided by” and that “per” is only used for a quantity of “one”, so you can say “per ...
Minkowski space
In mathematical physics, Minkowski space or Minkowski spacetime is a combination of Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded. Although initially developed by mathematician Hermann Minkowski for Maxwell's equations of electromagnetism, the mathematical structure of Minkowski spacetime was shown to be an immediate consequence of the postulates of special relativity.Minkowski space is closely associated with Einstein's theory of special relativity, and is the most common mathematical structure on which special relativity is formulated. While the individual components in Euclidean space and time will often differ due to length contraction and time dilation, in Minkowski spacetime, all frames of reference will agree on the total distance in spacetime between events. Because it treats time differently than the three spacial dimensions, Minkowski space differs from four-dimensional Euclidean space.The isometry group, preserving Euclidean distances of a Euclidean space equipped with the regular inner product is the Euclidean group. The analogous isometry group for Minkowski apace, preserving intervals of spacetime equipped with the associated non-positive definite bilinear form (here called the Minkowski inner product,) is the Poincaré group. The Minkowski inner product is defined as to yield the spacetime interval between two events when given their coordinate difference vector as argument.