Expression 7: "V" left parenthesis, "r" , "t" , right parenthesis equals StartFraction, Gamma Over 2 pi "r" , EndFraction left parenthesis, 1 minus exp left parenthesis, negative StartFraction, "r" squared Over 4 "v" "t" , EndFraction , right parenthesis , right parenthesisVr,t=Γ2πr1−exp−r24vt
7
Note: This is equal to the integral of V(r,t) dt, but I did some parts manually to speed things up. (The integral of the remaining parts is a special function not supported by Desmos)
8
Expression 9: theta Subscript, 1 , Baseline left parenthesis, "r" , "t" , right parenthesis equals StartFraction, Gamma Over 2 pi "r" , EndFraction "t" minus Start integral from 0 to "t" , end integral, "e" Superscript, negative StartFraction, "r" squared Over 4 "v" "t" , EndFraction , Baseline "d" "t" Subscript, 1 , Baseline StartFraction, 1 Over "r" , EndFractionθ1r,t=Γ2πrt−∫t0e−r24vtdt11r
9
Expression 10: "x" Subscript, 1 , Baseline left parenthesis, "r" , "t" , right parenthesis equals "r" cosine left parenthesis, theta Subscript, 1 , Baseline left parenthesis, "r" , "t" , right parenthesis , right parenthesisx1r,t=rcosθ1r,t
10
Expression 11: "y" Subscript, 1 , Baseline left parenthesis, "r" , "t" , right parenthesis equals "r" sine left parenthesis, theta Subscript, 1 , Baseline left parenthesis, "r" , "t" , right parenthesis , right parenthesisy1r,t=rsinθ1r,t
11
Expression 12: left parenthesis, "x" Subscript, 1 , Baseline left parenthesis, "t" , "T" , right parenthesis , "y" Subscript, 1 , Baseline left parenthesis, "t" , "T" , right parenthesis , right parenthesisx1t,T,y1t,T
domain t Minimum: 00
less than or equal to "t" less than or equal to≤t≤
domain t Maximum: 11
12
Hidden Label: left parenthesis, "x" Subscript, 1 , Baseline left parenthesis, left bracket, 0 , "a" ...4 0 "a" , right bracket , "T" , right parenthesis , "y" Subscript, 1 , Baseline left parenthesis, left bracket, 0 , "a" ...4 0 "a" , right bracket , "T" , right parenthesis , right parenthesis Has graph. To audio trace, press ALT+T.x10,a...40a,T,y10,a...40a,T