max and min of directed line segment

Question

Let F= i + j. 

How would you place a directed line segment C of length one so that the value of ∫_cF*dr would be

a. a maximum

b. a minimum

c. zero

d. what would the max and min values of the integral be?

Answer 1

I'll assume that F = i + j defines a 2D vector field that is constant with position (x,y),

and that i and j are the unit vectors along the x and y axes respectively.

So at any point (x,y), F has length √2 and points in a direction 45 degress from the x-axis, that is, parallel to the line y=x.

 

Now we have C which is a line segment of length 1 and points in some direction.

The line integral is essetially the sum of the dot product between the vectors F and dr as we go along the curce C:

    ∫_cF•dr  

When F and dr are in the same direction we get a maximum contribution, when they are opposite we get a minimum (large negative value) contribution, and if F and dr are perpendicular then we get 0 contribution from the dot product.

 

So for this particular, constant, field:

 

a) maximum when C is pointing in the direction i + j .  (45 degrees from X-axis)

 

b) a minimum when C is in the opposite direction:  -i + -j .  (225 degrees from X-axis)

 

c) 0 when C is pointing perpendicular to i + j, so in either direction:  i - j  or  -i + j  (+135 or -45 degress from X-axis)

 

d) Since F has magnitude √2 and C has length 1, we get max/min values of +/- √2 .