On Euler's formula... "а(φ) = cosφ + jsinφ; j^2 = -1...."

[Desert Fox]

On Euler's formula... "а(φ) = cosφ + jsinφ; j^2 = -1...."

Hello everybody,

I am dealing with a text, where I found the following notes about Euler's formula:

а(φ) = cosφ + jsinφ; j^2 = -1 ->
da(φ)/dφ = d(cosφ + jsinφ)/dφ = -sinφ + jcosφ = j(cosφ + jsinφ) = jа(φ) ->
∫[da(φ)/а(φ)] = j∫dφ ->
ln|а(φ)| = jφ + k ->
а(φ) = Ke^jφ; K = a(0) = cos0 + jsin0 = 1 ->
e^jφ = cosφ + jsinφ ->
A◦(t) = A(t)e^j[ω(t)t+ψ] = A(t)cos[ω(t)t+ψ] + jA(t)sin[ω(t)t+ψ] – a “circulating” vector;
φ - angle; ψ – “initial” angle; ω – angular velocity; t - time.

This is extremely complex notation for me. I need help from good Mathematician who can review the fragment and tell me whether there are errors in the quoted notation. And eventually.... give me a short, common sense answer to the question "what is meant by this impenetrable notation?"

Thank you very much for the efforts...

 

[stapel]

I am dealing with a text, where I found the following notes about Euler's formula:

а(φ) = cosφ + jsinφ; j^2 = -1 ->
da(φ)/dφ = d(cosφ + jsinφ)/dφ = -sinφ + jcosφ = j(cosφ + jsinφ) = jа(φ) ->
∫[da(φ)/а(φ)] = j∫dφ ->
ln|а(φ)| = jφ + k ->
а(φ) = Ke^jφ; K = a(0) = cos0 + jsin0 = 1 ->
e^jφ = cosφ + jsinφ ->
A◦(t) = A(t)e^j[ω(t)t+ψ] = A(t)cos[ω(t)t+ψ] + jA(t)sin[ω(t)t+ψ] – a “circulating” vector;
φ - angle; ψ – “initial” angle; ω – angular velocity; t - time.

This is extremely complex notation for me. I need help from good Mathematician who can review the fragment and tell me whether there are errors in the quoted notation. And eventually.... give me a short, common sense answer to the question "what is meant by this impenetrable notation?"

Um... I don't think there exists a "short" explanation of differential and integral calculus, or even just of its notation. Are you not familiar with these topics? Because, if not, the first step would probably be to spend a year or so studying them....