15.1 Vector Differentiation


1-7. Sketch the vector field by drawing a diagram.

1. .  

http://matrix.skku.ac.kr/cal-lab/cal-14-1-1.html 

 Sol)

 

var('x,y') 

vf=plot_vector_field((3,4), (x,-3,3), (y,-3,3), aspect_ratio=1);

show(vf)

 


2.

 http://matrix.skku.ac.kr/cal-lab/cal-15-1-2.html

 Sol)

 

var('x,y') 

vf=plot_vector_field((1/2*x,-2*y), (x,-3,3), (y,-3,3), aspect_ratio=1);

show(vf)


 


3. .  

 http://matrix.skku.ac.kr/cal-lab/cal-15-1-3.html

 Sol)

 

var('x,y') 

vf=plot_vector_field((5*y,1/2), (x,-3,3), (y,-3,3), aspect_ratio=1);

show(vf) 

 


4. .  

 http://matrix.skku.ac.kr/cal-lab/cal-15-1-4.html

 Sol)

 

var('x,y') 

vf=plot_vector_field((x+y,x), (x,-3,3), (y,-3,3), aspect_ratio=1);

show(vf) 


 


5. .

 http://matrix.skku.ac.kr/cal-lab/cal-15-1-5.html

 Sol)

 

var('x,y') 

vf=plot_vector_field((y/sqrt(x^2+y^2),-x/sqrt(x^2+y^2)), (x,-3,3), (y,-3,3), aspect_ratio=1);

show(vf) 


 


6. .  

 http://matrix.skku.ac.kr/cal-lab/cal-15-1-6.html

 Sol)

 

var('x,y') 

vf=plot_vector_field((y/sqrt(x^2+y^2),x/sqrt(x^2+y^2)), (x,-3,3), (y,-3,3), aspect_ratio=1);

show(vf) 


 


7. .  

 http://matrix.skku.ac.kr/cal-lab/cal-15-1-7.html

 Sol)

 

x,y,z=var('x,y,z') 

plot_vector_field3d((0,0,1), (x, -3,3), (y,-3,3), (z,-3,3))


 


8. Find (a) the curl and (b) the divergence of the vector field.

.

 Sol) (a) curl : ,  (b) div :    


.

 Sol) (a) curl : ,  (b) div : .


.

 Sol) (a) curl : ,  (b) div : .


.

 Sol) (a) curl : ,  (b) div : .


.

 Sol) (a) curl : ,  (b) div : .


.

 Sol) (a) curl : ,

     (b) div : .


.

 Sol) (a) curl : ,  (b) div : .


.

 Sol) (a) curl : ,  (b) div : .


9. Let be a scalar function and a vector field. Stats whether each expression is meaningful. If not, explain why. if so, state whether it is a scalar function or a vector field.

(a) curl

 Sol) not meaningful , curl must take a vector field.

(b) grad

 Sol) meaningful, vector field

(c) div

 Sol) meaningful, scalar function

(d) curl(grad)

 Sol) meaningful, vector field

(e) grad

 Sol) not meaningful

(f) grad(div)

 Sol) meaningful, vector field

(g) div(grad)

 Sol) meaningful, scalar function

(h) grad(div)

 Sol) not meaningful

(i) curl(grad)

 Sol) not meaningful

(j) div(div)

 Sol) not meaningful

(k) (grad)(div)

 Sol) not meaningful

(l) div(curl(grad))

 Sol) meaningful, scalar function


10. Is there a vector field on such that curl ? Explain.

 Sol) No, curl.


11. Is there a vector field on such that curl ? Explain.

 Sol) No, curl.


12-17. Determine whether or not the vector field is conservative. If it is conservative, find a potential of .


12. .

 Sol) not conservative since curl.


13. .

 Sol) not conservative


14. .

 Sol) not conservative


15. .

 Sol) conservative, Let .

     Then

          

            

          Therefore, .


16. .

 Sol) not conservative


17. .

 Sol) conservative , .


18. (a) Let be a differentiable vector field with

        div. Define a vector field by

       , ,

       and . Prove that .

     (b) Let . Find such that .


19. Find unit tangent vector to the curve , , at any time .

http://matrix.skku.ac.kr/cal-lab/cal-14-1-19.html

 Sol) 

 

x,y,z,t,u=var('x,y,z,t,u');

x(t)=4*sin(-t);

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

z(t)=5*t;          

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

r(t)

  (4*sin(-t), 2*cos(2*t), 5*t)


dr(t)=(diff(x(t),t),diff(y(t),t),diff(z(t),t)); # tangent vector(dr/dt)

dr(t) 

  (-4*cos(-t), -4*sin(2*t), 5)


dr(t)/abs(dr(t))+r(t) # unit tangent vector((dr/dt)/abs(dr/dt))

  (-4*cos(-t)/sqrt(16*sin(2*t)^2 + 16*cos(-t)^2 + 25) + 4*sin(-t),

  -4*sin(2*t)/sqrt(16*sin(2*t)^2 + 16*cos(-t)^2 + 25) + 2*cos(2*t),

  5*t + 5/sqrt(16*sin(2*t)^2 + 16*cos(-t)^2 + 25))


 And we can draw a curve with given unit tangent vector.

 

t0=2;

p1=r(t0);

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

p=parametric_plot3d((x(t),y(t),z(t)),(t,0,2*pi));

utv=line3d([p1,p2],rgbcolor=(1,0,0),arrow_head=True, thickness=5);

show(p+utv)

 

          


20. Find the directional derivative of at in the direction of .

http://matrix.skku.ac.kr/cal-lab/cal-15-1-20.html

 Sol) 

 

x,y,z=var('x,y,z');

f(x,y,z)=x*exp(y*z);

f(x,y,z) 

  x*e^(y*z)

 

df(x,y,z)=(diff(f,x),diff(f,y),diff(f,z)); # gradient of f

df(x,y,z)

  (e^(y*z), x*z*e^(y*z), x*y*e^(y*z))


df_p=vector(QQ, df(2,4,0));

df_p 

  (1, 0, 8)


v=vector(QQ, [1,-3,2]);

  (1, -3, 2)


u=v/abs(v);

  (1/14*sqrt(14), -3/14*sqrt(14), 1/7*sqrt(14))

 

duf=df_p*u;

duf 

  17/14*sqrt(14)


21. Find the equations of the tangent plane and normal line at the point to the paraboloid .

http://matrix.skku.ac.kr/cal-lab/cal-15-1-21.html

 Sol)

 

x,y,z,t=var('x,y,z,t');

F(x,y,z)=x^2+y^2+z;

F(x,y,z) 

  x^2 + y^2 + z


 

dF(x,y,z)=(diff(F,x),diff(F,y),diff(F,z));

dF(x,y,z) 

  (2*x, 2*y, 1)

 

dF_p=vector(QQ,dF(-1,1,-2));

dF_p 

  (-2, 2, 1)

 

p=vector(QQ,[-1,1,-2]);

w=vector(SR,[x,y,z]);

w-p 

  (x + 1, y - 1, z + 2)

 

dF_p*(w-p)==0

  -2*x + 2*y + z - 2 == 0


solve((x+1)/-2==t,x)

  [x == -2*t - 1]

 

solve((y-1)/2==t,y)

  [y == 2*t + 1]

 

solve((z+2)/1==t,z)

  [z == t - 2]

 

p1=implicit_plot3d(z==-x^2-y^2,(x,-7,7),(y,-7,7),(z,-7,7), opacity=0.2, color="red", mesh=True);

p2=implicit_plot3d(dF_p*(w-p)==0,(x,-7,7),(y,-7,7),(z,-7,7), opacity=0.2, color="blue",
mesh=True);

p3=parametric_plot3d((-2*t-1,2*t+1,t-2),(t,-1,1), opacity=1, color="red", mesh=True);

show(p1+p2+p3, aspect_ratio=1)


    


22. Find .

http://matrix.skku.ac.kr/cal-lab/cal-14-1-22.html

 Sol)

 

x,y,z,i,j,k=var('x,y,z,i,j,k')

P(x,y,z) = exp(x*z);

Q(x,y,z) = x*y^2*z;

R(x,y,z) = -y^2+z^2;

A(x,y,z) = P*i+Q*j+R*k;

A(x,y,z) 

  j*x*y^2*z - (y^2 - z^2)*k + i*e^(x*z)


 

divA = diff(P,x)+diff(Q,y)+diff(R,z)

divA(x,y,z) 

  2*x*y*z + z*e^(x*z) + 2*z


23. Show that the vector function is not conservative.

http://matrix.skku.ac.kr/cal-lab/cal-15-1-23.html

 Sol)

 

x,y,z,i,j,k=var('x,y,z,i,j,k')

P(x,y,z) = exp(x*z);

Q(x,y,z) = 3*x*y*z;

R(x,y,z) = 2*y;

A(x,y,z) = P*i+Q*j+R*k;

A(x,y,z)

  3*j*x*y*z + i*e^(x*z) + 2*k*y


 

curlA = (diff(R,y)-diff(Q,z))*i+(diff(P,z)-diff(R,x))*j+(diff(Q,x)-diff(P,y))*k;

curlA(x,y,z)

  j*x*e^(x*z) + 3*k*y*z - (3*x*y - 2)*i


 Since curl, is not conservative.


24. If , evaluate .

http://matrix.skku.ac.kr/cal-lab/cal-14-1-24.html

 Sol)

 

t=var('t')                     

r=vector([t,-t^2,t-1])

w=vector([2*t^2,0,6*t])

v=r.dot_product(w)

v.integrate(t,0,2)

  12



25. If , , then evaluate

    a. ,

    b. .


26. If , then evaluate .