13.2 Calculus of Vector Functions

1-5. Find and draw the position vector and the tangent vector for the given value of .


1.

http://matrix.skku.ac.kr/cal-lab/cal-13-2-1.html

 Sol)

  

var('t');

x(t)=cos(t);

y(t)=t;

r(t)=(x(t),y(t));

dr(t)=(diff(x(t),t),diff(y(t),t));

dr(t)

  (-sin(t),1)


 So, we can see that .

  

t0=pi/4;

p1=r(t0);

p2=dr(t0)/abs(dr(t0))+r(t0);

p=parametric_plot((x(t),y(t)),(t,0,2*pi));

utv=line([p1,p2],rgbcolor=(1,0,0),thickness=2);

show(p+utv);


 


2. , .

http://matrix.skku.ac.kr/cal-lab/cal-13-2-2.html

 Sol)

  

var('t');

x(t)=sin(t);y(t)=t;

r(t)=(x(t),y(t));

dr(t)=(diff(x(t),t),diff(y(t),t));dr(t)

  (cos(t),1)


 So, we can see that .

  

t0=0;

p1=r(t0);

p2=dr(t0)/abs(dr(t0))+r(t0);

p=parametric_plot((x(t),y(t)),(t,0,2*pi));

utv=line([p1,p2],rgbcolor=(1,0,0),thickness=2);

show(p+utv);


 


3. , .

http://matrix.skku.ac.kr/cal-lab/cal-13-2-3.html

 Sol)

  

var('t');

x(t)=t^3;y(t)=t^2;

r(t)=(x(t),y(t));

dr(t)=(diff(x(t),t),diff(y(t),t));dr(t)

  (3*t^2, 2*t)


 So, we can see that .

  

t0=1;

p1=r(t0);

p2=dr(t0)/abs(dr(t0))+r(t0);

p=parametric_plot((x(t),y(t)),(t,0,2));

utv=line([p1,p2],rgbcolor=(1,0,0),thickness=2);

show(p+utv);


 


4. , .

http://matrix.skku.ac.kr/cal-lab/cal-13-2-4.html

 Sol)

  

var('t');

x(t)=exp(2*t);

y(t)=exp(-3*t);

r(t)=(x(t),y(t));

dr(t)=(diff(x(t),t),diff(y(t),t));

dr(t)

  (2*exp(2*t), -3*exp(-3*t))


 So, we can see that .

  

t0=0;

p1=r(t0);

p2=dr(t0)/abs(dr(t0))+r(t0);

p=parametric_plot((x(t),y(t)),(t,-2,2));

utv=line([p1,p2],rgbcolor=(1,0,0),thickness=2);

show(p+utv);

 


5. , .

http://matrix.skku.ac.kr/cal-lab/cal-13-2-5.html

 Sol)

  

var('t');

x(t)=3*sin(t);

y(t)=2*cos(t);

r(t)=(x(t),y(t));

dr(t)=(diff(x(t),t),diff(y(t),t));

dr(t)

  (3*cos(t), -2*sin(t))


 So, we can see that .

  

t0=pi/3;

p1=r(t0);

p2=dr(t0)/abs(dr(t0))+r(t0);

p=parametric_plot((x(t),y(t)),(t,-2*pi,2*pi));

utv=line([p1,p2],rgbcolor=(1,0,0),thickness=2);

show(p+utv);


 


6-10. Find the derivatives of the vector functions.


6. .

 Sol) .


7. .

 Sol)


8. .

 Sol) .


9. .

 Sol) .


10. .

 Sol) .


11-13. Find the unit tangent vector at the point with the given value of the parameter .


11. , .

 Sol) ,

      ,

     Hence .


12. , .

 Sol) .

      .

     Therefore, .


13. , .

 Sol) ,

      ,

      .


14. If , find , , , and .

 Sol) , .

     Therefore, ,

         .

         .


15. If , find and .

 Sol) .

     .

     .

     .

     .

     .


16-19. Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point.


16. ; .

 Sol) We can see that .

     For , . So, .

     Hence .


17. ; .

 Sol) We can see that .

     For , .

     Therefore, .

     Hence .


18. , , ; .

 Sol) We can see that .

     For , . So, .

     Hence .


19. , , ; .

 Sol) We can see that .

     For , . So, .

     Hence .


20-21. Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. Illustrate by graphing both the curve and the tangent line together.


20. , , ; .

http://matrix.skku.ac.kr/cal-lab/cal-13-2-20.html

 Sol) We can see that .

  

var('t,s')

x(t)=t; y(t)=2*cos(t); z(t)=2*sin(t);

r(t)=(x(t),y(t),z(t));

dr(t)=(diff(x(t),t),diff(y(t),t),diff(z(t),t));

t0=pi/3;

pe=r(t0)+dr(t0)*s;pe

  (1/3*pi + s, -sqrt(3)*s + 1, s + sqrt(3))


 So, parametric equation is .

  

p1=parametric_plot3d(r(t),(t,-pi,pi));

p2=parametric_plot3d(pe,(s,-pi,pi),color='red);

show(p1+p2);


 


21. , , ; .

http://matrix.skku.ac.kr/cal-lab/cal-13-2-21.html

 Sol) We can see that .

  

var('t,s')

x(t)=exp(t); y(t)=sin(t); z(t)=2*exp(-3*t);

r(t)=(x(t),y(t),z(t));

dr(t)=(diff(x(t),t),diff(y(t),t),diff(z(t),t));

t0=0;

pe=r(t0)+dr(t0)*s;pe

 

  (1 + s, s, 2 -6*s)


 So, parametric equation is .

  

p1=parametric_plot3d(r(t),(t,-2,2));

p2=parametric_plot3d(pe,(s,-2,2),color='red);

show(p1+p2);

 


22-24. Determine whether the curve are smooth.


22. .

 Sol) . Since , this curve is not smooth.


23. .

 Sol) .

     First coordinate of cannot be zero for any real number , so this curve is smooth.


24. .

 Sol) .

     Second coordinate of cannot be zero for any real number , so this curve is smooth.


25-30. Evaluate the integrals.

25. .

 Sol) .


26. .

 Sol) .


27. .

 Sol) .


28. .

 Sol) where is a constant vector.


29. .

 Sol) where is a constant vector.


30. .

 Sol) .


31. Find if and .

 Sol) , where is a constant vector.

      Since , we finally have .


32. Find if and .

 Sol) , where is a constant vector.

      Since , .

      So, we finally have .


33. If and , use Theorem 4(Rules of differentiation) to find

 Sol) , .

     , .

     So, .


34. If and , use Theorem 2(Rules of differentiation) to find .

 Sol) ,

     ,

     ,

     .


35. Show that if is a vector function such that exists, then .

 Proof) Using Theorem 4(Rules of differentiation),

       .


36. If , show that .

 Proof) We already know that .

       Differentiate both sides, .

       So, we finally have .