Journey To Learn The Glorious Maxwell Equations_3_Math Tools

I encountered a big obstacle in the journey of grasping Maxwell Equations because there are some math tools I did’t understand yet.

Concepts such as scalar, vector, vector’s multiplication, gradient, partial differentiation etc. So I need to get into these math first.

Chanwei Jun provided the best lectures I ever have read to explain these abstract concepts so clearly.

Scalars are values without directions such as mass, weight, height, we’re familiar with vector from math course in high schools. Vectors are values with directions such as velocity, electric field. One cannot describe it without indicating the direction hat.

The math operators – plus, minus, multiplication have different behaviors between scalars and vectors.

vector plus a vector, for example, in below graph, vector OA (y axis) + vector OB (x axis) = OC (diagonal line)

vector dot vector, OA·OB=|OA||OB|Cosθ, the strict expression is (x1x+y1y).(x2x, y2y) = x1x2xx+x1y2xy+y1x2yx+y1y2yy, note xx = 1 because scalar unit 1 time itself is 1, xy = 0, because cosxy = cos90=0, so the upshot is  x1x2 +y1y2.

vector time vector OA x OB = |OA||OB|Sinθ.

The visual of dot and time product can be seen as a and b respectively

In a three-dimensional world, we can have an independent value z depended on two values x and y, it’s visually understandable if we think of z is the height of a mountain as shown below:

the math way is z = f(x, y), to get the differentiation view, dz = df/d?,? indicates the problem the mathematicians boggled down temporarily, and then they came up with a solution – partial differentiation on x and y respectively, i.e.

The insightful mathematician/physicists can tell it’s a dot result from two vectors dot product:

the partial differentiation on x and y can be symbolized as delz, (dxx + dyy) is familiar vector summation, symbolized as dl, the diagonal line from mountain sliding downward, then the equation can be written as

According to the definition of vector dot from another math angle we know

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