In[45]:= Mathematica TutorialIn[46]:= 1. Plot the graph of the function f Hx, yL = x2 + y2 on the region @-3, 3D�@-3, 3DIn[46]:= Plot3DAx2 + y2, 8x, -3, 3<, 8y, -3, 3<E

Out[46]=

-2

0

2 -2

0

2

0

5

10

15

2 Mathematica tutorial project 1.nb

In[47]:= 2. Plot the graph of the functions f Hx, yL =

x2 + y2 and g Hx, yL = x3 - y3 on the region @-3, 3D�@-3, 3D on the same graph

In[47]:= Plot3D@8x^2 + y^2, x^3 - y^3<, 8x, -3, 3<, 8y, -3, 3<D

Out[47]=

-2

0

2

-20

2

-20

0

20

Mathematica tutorial project 1.nb 3

4 Mathematica tutorial project 1.nb

In[48]:= 3. Plot the contour plot of the x2 + y2 = 1

Set::write : Tag Times in 3. Icontour of plot Plot the2

x2

+ y2M is Protected. �

Out[48]= 1

In[49]:= ContourPlot@x^2 + y^2 � 1, 8x, -3, 3<, 8y, -3, 3<D

Out[49]=

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

Mathematica tutorial project 1.nb 5

6 Mathematica tutorial project 1.nb

In[50]:= 3. Plot the graph of the curve r® HtL = < cos H3 tL, sin H3 tL, sin H2 tL > without the axis

In[50]:= ParametricPlot3D@8Cos@3 tD, Sin@3 tD, Sin@2 tD<, 8t, 0, 2 Pi<, Boxed ® False, Axes ® NoneD

Out[50]=

Mathematica tutorial project 1.nb 7

8 Mathematica tutorial project 1.nb

In[51]:= 4. Find the velocity and acceleration vectors for the r® HtL = < cos H3 tL, sin H3 tL, sin H2 tL >

In[51]:= r = 8Cos@3 tD, Sin@3 tD, Sin@2 tD<velocity = D@r, tDacceleration = D@velocity, tD

Out[51]= 8Cos@3 tD, Sin@3 tD, Sin@2 tD<

Out[52]= 8-3 Sin@3 tD, 3 Cos@3 tD, 2 Cos@2 tD<

Out[53]= 8-9 Cos@3 tD, -9 Sin@3 tD, -4 Sin@2 tD<

Mathematica tutorial project 1.nb 9

10 Mathematica tutorial project 1.nb

5. Find the unit tangent and the normal vectors for the r® HtL = < cos H3 tL, sin H3 tL, sin H2 tL >

In[54]:= T = SimplifyB velocity

velocity.velocity

F

NormalVector = SimplifyB D@T, tDD@T, tD.D@T, tD

F

Out[54]= :-

3 Sin@3 tD

11 + 2 Cos@4 tD,

3 Cos@3 tD

11 + 2 Cos@4 tD,

2 Cos@2 tD

11 + 2 Cos@4 tD>

Out[55]= :-

5 Cos@tD + 33 Cos@3 tD + Cos@7 tD

H11 + 2 Cos@4 tDL3�2 107+10 Cos@4 tDH11+2 Cos@4 tDL2

,

-

-5 Sin@tD + 33 Sin@3 tD + Sin@7 tD

H11 + 2 Cos@4 tDL3�2 107+10 Cos@4 tDH11+2 Cos@4 tDL2

, -

12 Sin@2 tD

H11 + 2 Cos@4 tDL3�2 107+10 Cos@4 tDH11+2 Cos@4 tDL2

>

In[56]:=

Mathematica tutorial project 1.nb 11

12 Mathematica tutorial project 1.nb

6. Find the curvature of the curve r® HtL = < cos H3 tL, sin H3 tL, sin H2 tL >

In[57]:= Κ =1

velocity.velocity

K D@T, tD.D@T, tD O �� Simplify

Out[57]=

3107+10 Cos@4 tDH11+2 Cos@4 tDL2

11 + 2 Cos@4 tD

Mathematica tutorial project 1.nb 13

14 Mathematica tutorial project 1.nb

7. Compute the curvature at t =

3 and approximate it value to 10 digits after the decimal point

Out[58]= 30 after and approximate decimal digits it point the to value

In[59]:= curv@t_D :=

3107+10 Cos@4 tDH11+2 Cos@4 tDL2

11 + 2 Cos@4 tDcurv@3DN@curv@3D, 10D

Out[60]=3 107 + 10 Cos@12DH11 + 2 Cos@12DL3�2

Out[61]= 0.7132173645

Mathematica tutorial project 1.nb 15

16 Mathematica tutorial project 1.nb

8. Find the length of the curve r® HtL = < cos H3 tL, sin H3 tL, sin H2 tL > from t = 0 to t = 1

In[67]:= l = NBà0

1

velocity.velocity âtFOut[67]= 3.25162

Mathematica tutorial project 1.nb 17

18 Mathematica tutorial project 1.nb

9. Find the arclength parametrization of the curve r® HtL = < t, t4 >

In[94]:= c = 83 t, 5 t<v = D@c, tDv.v �� Simplify

s = à0

u

34 ât

Clear@sDOut[94]= 83 t, 5 t<

Out[95]= 83, 5<

Out[96]= 34

Out[97]= 34 u

In[99]:= SolveBs == 34 u, uFOut[99]= ::u ®

s

34>>

Mathematica tutorial project 1.nb 19