212HomeWork1sol - Physics 212A Homework#1 Name 1 Practice...

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Physics 212A Homework #1 Name: 1. Practice in transcribing expressions into Mathematica syntax a) Find the determinant of this matrix: ( use Insert, Table/Matrix ) Type it in using Insert Table as per suggestion In[14]:= mat1 = 1 2 3 4 1 - 1 0 1 - 1 2 6 7 0 2 4 1 Out[14]= 88 1, 2, 3, 4 < , 8 1, - 1, 0, 1 < , 8 - 1, 2, 6, 7 < , 8 0, 2, 4, 1 << then compute Det In[15]:= Det @ mat1 D Out[15]= 57 First, try to compute y In[16]:= y = Integrate @ Sin @ Sqrt @ t^3 + 1 DD , 8 t, 0, Sqrt @ x D<D Out[16]= 0 x Sin B 1 + t 3 F t It can' t do it. But it can correctly calculate the derivative
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In[17]:= D @ y, x D Out[17]= Sin B 1 + x 3 2 F 2 x c) Find the first 3 terms in the Taylor's series expansion of about x=0. In[18]:= Series @ Sqrt @ a + Log @ x + 1 DD , 8 x, 0, 3 <D Out[18]= a + x 2 a + H - 1 - 2a L x 2 8a 3 2 + I 3 + 6a + 8a 2 M x 3 48a 5 2 + O @ x D 4 d) Let A= 1 2 5 - 2 4 1 0 - 1 3 and B= 1 1 1 - 2 6 0 4 2 7 . Compute the comutator AB-BA and verify that H AB L T = B T A T and that H A B L - 1 = B - 1 A - 1 Type the matrices in the old fashioed way : In[19]:= A = 88 1, 2, 5 < , 8 - 2, 4, 1 < , 8 0, - 1, 3 << Out[19]= 88 1, 2, 5 < , 8 - 2, 4, 1 < , 8 0, - 1, 3 << In[20]:= B = 88 1, 1, 1 < , 8 - 2, 6, 0 < , 8 4, 2, 7 << Out[20]= 88 1, 1, 1 < , 8 - 2, 6, 0 < , 8 4, 2, 7 << Use MatrixForm to make sure we got it right : In[23]:= MatrixForm @ A D Out[23]//MatrixForm= 1 2 5 - 2 4 1 0 - 1 3 In[24]:= MatrixForm @ B D Out[24]//MatrixForm= 1 1 1 - 2 6 0 4 2 7 Commutator = In[25]:= A.B - B.A Out[25]= 88 18, 18, 27 < , 8 8, 4, 9 < , 8 14, - 9, - 22 << 2 212HomeWork1sol.nb
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In[26]:= MatrixForm @ % D Out[26]//MatrixForm= 18 18 27 8 4 9 14 - 9 - 22 H AB L T = B T A T : In[27]:= TrAB = Transpose @ A.B D Out[27]= 88 17, - 6, 14 < , 8 23, 24, 0 < , 8 36, 5, 21 << In[28]:= TrBTrA = Transpose @ B D .Transpose @ A D Out[28]= 88 17, - 6, 14 < , 8 23, 24, 0 < , 8 36, 5, 21 << In[29]:= TrAB - TrBTrA Out[29]= 88 0, 0, 0 < , 8 0, 0, 0 < , 8 0, 0, 0 << In[30]:= TrAB TrBTrA Out[30]= True H A B L - 1 = B - 1 A - 1 : In[31]:= Inverse @ A.B D - Inverse @ B D .Inverse @ A D Out[31]= 88 0, 0, 0 < , 8 0, 0, 0 < , 8 0, 0, 0 << 212HomeWork1sol.nb 3
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In[32]:= Inverse @ A.B D == Inverse @ B D .Inverse @ A D Out[32]= True Use Mathematica to evaluate e) x Tan @ k x D 2 x In[33]:= Integrate @ xTan @ kx D ^2, x D Out[33]= - x 2 2 + Log @ Cos @ kx DD k 2 + xTan @ kx D k f) 0 ¥ e - s x x 3 Cos @ k x D x for positive s and real k. In[34]:= Integrate A ª - sx x 3 Cos @ kx D , 8 x, 0, ¥ < , Assumptions fi 8 s > 0, k ˛ Reals <E Out[34]= 6 I k 4 - 6k 2 s 2 + s 4 M I k 2 + s 2 M 4 g) 0 ¥ J 0 @ k r D 1 + k 2 r 2 r for positive values of k (look up BesselJ ) In[35]:= Integrate B BesselJ @ 0, kr D 1 + k 2 r 2 , 8 r, 0, ¥ < , Assumptions fi k > 0 F Out[35]= BesselI A 0, 1 2 E BesselK A 0, 1 2 E k 2. Practice using DSolve. The differential equation d n d t = Α n - Β n 2 , is a model for population dynamics. The number of individuals n is a function of t, while Α and Β are constants. a) Find the general solution to this differential equation. In[37]:= n1 = n @ t D . DSolve @ n' @ t D Α * n @ t D - Β * n @ t D ^2, n @ t D , t D Out[37]= : ª t Α Α Α C @ 1 D + ª t Α Β > If the curly brackets are irritating, get rid of themwith either First of Flatten In[38]:= n1 = n @ t D . First @ DSolve @ n' @ t D Α * n @ t D - Β * n @ t D ^2, n @ t D , t DD Out[38]= ª t Α Α Α C @ 1 D + ª t Α Β 4 212HomeWork1sol.nb
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In[39]:= n1 = n @ t D . Flatten @ DSolve @ n' @ t D Α * n @ t D - Β * n @ t D ^2, n @ t D , t DD Out[39]= ª t Α Α Α C @ 1 D + ª t Α Β b)The population approaches a constant as t fi¥ . What is the constant asymptotic population? Use Limit and Assumptions.
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