Find the vector B from the origin to the opposite corner that lies in the xy plane

a

x

y z

A

B

2. Find the vector B from the origin to the opposite corner that lies in the xy plane.

3^MATLAB. Given two vectors A = 3ux + 4uy + 5uz and B = -5ux + 4uy - 3uz, find C

= A + B and D = A - B. In addition, carefully illustrate these vectors using MATLAB.

4^MATLAB. Using the vectors defined in 3, evaluate A x B and A x B. Check your answer with MATLAB.

5^MATLAB. Given two vectors A = ux + uy + uz and B = 2ux + 4uy + 6uz, find C = A + B and D = A - B. In addition, carefully illustrate these vectors using MATLAB.

6^MATLAB. Using the vectors defined in 5, evaluate A x B and B x A. Check your answer with MATLAB.

7^MATLAB. Using MATLAB, write a program to convert degrees C to degrees F. Plot the results.

8^MATLAB. Using MATLAB, write a program to convert a yard stick to a meter stick.

Plot the results.

9^MATLAB. Using MATLAB, plot y = e^-x on a linear and a semilog graph.

10^MATLAB. Using MATLAB, plot two cycles of y = cos(x) on a linear and a polar graph.

11^MATLAB. Using MATLAB, carefully plot a vector field defined by A = y2ux - xuy in the region - 2 < x < + 2, - 2 < y < + 2. The length of the vectors in the field should be proportional to the field at that point. Find the magnitude of this vector at the point (3, 2).

12^MATLAB. Using MATLAB, carefully plot a vector field defined by in the

A = sin x ux – sin y uy in the region 0 < x < S, 0 < y < S. The length of the vectors in the field should be proportional to the field at that point. Find the

magnitude of this vector at the point ( 2 S,

2 S).

13^MATLAB. Find the scalar product of the two vectors defined by

A = 3_ux + 4_uy + 5_uz and ^B = -5_ux + 4_uy - 3_uz. Determine the angle between these two vectors. Check your answer using MATLAB.

14^MATLAB. Find the scalar product of the two vectors defined by

A = _ux + _uy + _uz and ^B = 2_ux + 4_uy + 6_uz. Determine the angle between these two vectors. Check your answer using MATLAB.

15^MATLAB. Find the projecthetion of a vector from the origin to a point defined at (1,2,3) on the vector from the origin to a point defined at (2,1,6). Find the angle between these two vectors. Check your answer using MATLAB.

16^MATLAB. Find the vector product of the two vectors defined by

A = 3_ux + 4_uy + 5_uz and ^B = -5_ux + 4_uy - 3_uz. Check your answer using MATLAB.

17^MATLAB. Find the vector product of the two vectors defined by

A = _ux + _uy + _uz and ^B = 2_ux + 4_uy + 6_uz. Check your answer using MATLAB.

18^MATLAB. Express the vector field A = 3_ux + 4_uy + 5_uz in cylindrical coordinates.

Check your answer using MATLAB.

19^MATLAB. Express the vector B = 3_ur + 4u_I + 5_uz that is in cylindrical coordinates into Cartesian coordinates. Check your answer using MATLAB.

20^MATLAB. Express the vector field A = 3_ux + 4_uy + 5_uz in spherical coordinates.

Check your answer using MATLAB.

21^MATLAB. Express the vector B = 3uU + 4u_T + 5u_I that is in spherical coordinates into Cartesian coordinates. Check your answer using MATLAB.

22^MATLAB. For the vectors A = _ux + _uy + _uz, B = 2_ux + 2_uy + 2_uz, and

C = 3_ux +3_uy + 3_uz; show that A x (B x C) = B(A • C) - C(A • B). Check your answer using MATLAB.

23^MATLAB. For the vectors A = _ux + 3_uy + 5_uz, B = 2_ux + 4_uy + 6_uz, and

C = 3_ux +4_uy + 5_uz; show that A x (B x C) = B(A • C) - C(A • B). Check your answer using MATLAB.

24^MATLAB. Find the area of the parallelogram using vector notation. Compare your result with that found graphically.

25^MATLAB. Show that we can use the vector definitions A • B = 0 and A x B = 0 to express that two vectors are perpendicular and parallel to each other respectively.

26^MATLAB. Let A = -2_ux + 3_uy + 4_uz; B = 7_ux + 1_uy + 2_uz; and

C = -1_ux + 2_uy + 4_uz. Find (a) A x B. (b) (A x B) • C. (c) A • (B x C).

27. Calculate the work required to move a mass m against a force field F = 5_ux + 7_uy along the indicated direct path from point a to point b.

5

10 x y

0 5

0 a

b

28. Calculate the work required to move a mass m against a force field F = y_ux + x_uy along the path abc and along the path adc. Is this field conservative?

5

10 x

0 5

0 a b

d c

29. Calculate the work required to move a mass m against a force field F = r_ur + rI^uI along the path abc.

y

-2 0 +2 x

a b

c

30. Calculate the work required to move a mass m against a force field F = rIu_I if the radius of the circle is a and 0 ” I ” 2ʌ.

31. Calculate the closed surface integral

³

^{A x ds if} A = xux + yuy and the surface is the surface of a cube.

Then apply divergence theorem to solve the same integral.

x

y z

2

32. Evaluate the closed surface integral of the vector

A = xyz ux + xyz uy + xyz uz over the cubical surface shown in Problem 31.

33 Evaluate the closed surface integral of the vector A = 3 u_U over the spherical surface that has a radius a.

34. Find the surface area of a cylindri-cal surface by setting up and evaluat-ing the integral

³

Axds ^where

z

r u

u

A 1 2 .

L a

35. A hill can be modeled with the equation H = 10 - x2 - 3y2 where H is the elevation of the hill. Find the path that a frictionless ball would take in order that it experienced the greatest change of elevation in the shortest change of horizontal position. Assume that the motion of the ball is unconstrained.

36. Find the gradient of the function H = x2yz and also the directional derivative of H specified by the unit vector u = a (ux + uy + uz) where a is a constant at the point (1, 2, 3). State the value for the constant a.

37. By direct differentiation show that ¸¸

¹

¨¨ ·

©

§

’ U

¸¸¹

¨¨ ·

©

§

’ U 1

1 '

where



x  x'



² 



y  y'



² 



z  z'



²

U

and ’' denotes differentiation with respect to the variables x', y', and z'.

38. Calculate the divergence of the vector

A = x3y sin (Sz) ux + xy sin (Sz) uy + x2y2z2 uz at the point (1,1,1).

39. Show that the divergence theorem is valid for a cube located at the center of a Cartesian coordinate system for a vector

A = xux + 2uy.

x

y z

2a

40. Show that the divergence theorem is valid for a sphere of radius a located at the center of a coordinate system for a vector A = U u_U .

0. A small paddle wheel with its axis parallel to the z axis is inserted into the channel and is free to rotate. Find the relative rates of rotation at the points

¸¹

Will the paddle wheel rotate if its axis is parallel to the x axis or the y axis?

0

a x

y z

42. Evaluate the line integral of the vector function A = x ux + x2y uy + xyz uz around the square contour C. Integrate ’ x A over the surface bounded by C. Show that this example satisfies Stokes's theorem.

x

y

z

In document Parte 1 Fundamentos Electromagneticos Con Matlab_Lonngren & Savov (Page 90-97)