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chwala
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- TL;DR Summary
- I have just been looking at the integration of ##e^{-x^2}##.
My first point of reference is:
https://math.stackexchange.com/questions/154968/is-there-really-no-way-to-integrate-e-x2
I have really taken time to understand how they arrived at ##dx dy=dA=r dθ dr## wow! I had earlier on gone round circles! ...i now get it that one is supposed to use partial derivatives
I managed to follow through the link here
https://math.stackexchange.com/questions/1636021/rigorous-proof-that-dx-dy-r-dr-d-theta
...but there is a slight mistake here: i.e on the line of
##dx dy = (\sin θ dr)(-r \sin θ dθ)-(\cos θ dr)( \cos θ dθ)##
##r## is missing!
It ought to be:
##dx dy = (\sin θ dr)(-r \sin θ dθ)-(\cos θ dr)(r \cos θ dθ)##.
In my approach i would have used the following lines,
Let ##x = r \cos θ## and ##y = r \sin θ##
and ##X=rθ## Where ##X## is a function of two variables, ##r## and ##θ##.
then,
##dx=x_r dr +x_θ dθ##
##dx=\cos θ dr -r \sin θ dθ ##
##dy=y_r dr +y_θ dθ##
##dy=\sin θ dr +r \cos θ dθ##
##dx dy = (\cos θ dr)(r \cos θ dθ)-(-r \sin θ dθ)(\sin θ dr)##
##dx dy = (\cos θ dr)(r \cos θ dθ)+(r \sin θ dθ)(\sin θ dr)##
...
Is there another way of looking at ##dA=dxdy##? Any insight guys...
My other question would be on the so called error function erf realised after integrating ##e^{-x^2}##. Any concrete reason as to why Mathematicians settled with the acronym erf? I understand that there are no trig/exponential substitutions that may be applicable on any other limits other than plus or minus infinity...cheers.
https://math.stackexchange.com/questions/154968/is-there-really-no-way-to-integrate-e-x2
I have really taken time to understand how they arrived at ##dx dy=dA=r dθ dr## wow! I had earlier on gone round circles! ...i now get it that one is supposed to use partial derivatives
I managed to follow through the link here
https://math.stackexchange.com/questions/1636021/rigorous-proof-that-dx-dy-r-dr-d-theta
...but there is a slight mistake here: i.e on the line of
##dx dy = (\sin θ dr)(-r \sin θ dθ)-(\cos θ dr)( \cos θ dθ)##
##r## is missing!
It ought to be:
##dx dy = (\sin θ dr)(-r \sin θ dθ)-(\cos θ dr)(r \cos θ dθ)##.
In my approach i would have used the following lines,
Let ##x = r \cos θ## and ##y = r \sin θ##
and ##X=rθ## Where ##X## is a function of two variables, ##r## and ##θ##.
then,
##dx=x_r dr +x_θ dθ##
##dx=\cos θ dr -r \sin θ dθ ##
##dy=y_r dr +y_θ dθ##
##dy=\sin θ dr +r \cos θ dθ##
##dx dy = (\cos θ dr)(r \cos θ dθ)-(-r \sin θ dθ)(\sin θ dr)##
##dx dy = (\cos θ dr)(r \cos θ dθ)+(r \sin θ dθ)(\sin θ dr)##
...
Is there another way of looking at ##dA=dxdy##? Any insight guys...
My other question would be on the so called error function erf realised after integrating ##e^{-x^2}##. Any concrete reason as to why Mathematicians settled with the acronym erf? I understand that there are no trig/exponential substitutions that may be applicable on any other limits other than plus or minus infinity...cheers.
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