Question

The figure shows a vector field\({\rm{F}}\)and two curves \({{\rm{C}}_{\rm{1}}}\)and\({{\rm{C}}_{\scriptstyle{\rm{2}}\atop\scriptstyle}}\). Are the line integrals of \({\rm{F}}\) over \({{\rm{C}}_{\rm{1}}}\)and \({{\rm{C}}_{\scriptstyle{\rm{2}}\atop\scriptstyle}}\)positive, negative, or zero?Explain.

Short answer

Line integral over \({{\rm{C}}_{\scriptstyle{\rm{1}}\atop\scriptstyle}}\)is positive.

Line integral over \({{\rm{C}}_{\scriptstyle{\rm{2}}\atop\scriptstyle}}\)is negative.

Step-by-step solution

Defining Integral over\({\rm{C}}\).

\(\begin{array}{c}\int_{\rm{C}} {\rm{F}} \cdot {\rm{dr = }}\int_{\rm{C}} {\rm{F}} \cdot {\rm{Tdt}}\\{\rm{ = }}\int_{\rm{C}} {\rm{|}} {\rm{F||T|cos\theta dt}}\end{array}\)

Where\(\theta \)is the angle between tangent and vector field

When \(\theta \)is acute, \({\rm{cos\theta }}\)is positive and it is negative \(\theta \)is obtuse.

Finding the line Integral.

The tangent vector makes an acute angle with the vector field at all points on\({{\rm{C}}_{\scriptstyle{\rm{1}}\atop\scriptstyle}}\).

Therefore, the line integral over \({{\rm{C}}_{\scriptstyle{\rm{1}}\atop\scriptstyle}}\)is positive.

The tangent vector makes an obtuse angle with the vector field at all points on\({{\rm{C}}_{\scriptstyle{\rm{2}}\atop\scriptstyle}}\).

Therefore, the line integral over \({{\rm{C}}_{\scriptstyle{\rm{2}}\atop\scriptstyle}}\)is negative.

Hence,

Line integral over \({{\rm{C}}_{\scriptstyle{\rm{1}}\atop\scriptstyle}}\)is positive and Line integral over \({{\rm{C}}_{\scriptstyle{\rm{2}}\atop\scriptstyle}}\)is negative.