Quiz (Fundamental Theorem of Line Integrals)

Show all work and justify each step.

(1) Show that the vector field fundamental theorem of line integrals quiz _gr_1.gif] is conservative and find a scalar potential fundamental theorem of line integrals quiz _gr_2.gif] for fundamental theorem of line integrals quiz _gr_3.gif] Then evaluate the line integral fundamental theorem of line integrals quiz _gr_4.gif] where fundamental theorem of line integrals quiz _gr_5.gif] is any smooth path connecting fundamental theorem of line integrals quiz _gr_6.gif] to fundamental theorem of line integrals quiz _gr_7.gif]

(2)  Show that the vector field

fundamental theorem of line integrals quiz _gr_8.gif]

is conservative and find a scalar potential fundamental theorem of line integrals quiz _gr_9.gif] for fundamental theorem of line integrals quiz _gr_10.gif]

(3) Show that the vector field

fundamental theorem of line integrals quiz _gr_11.gif]

is conservative and then evaluate the line integral fundamental theorem of line integrals quiz _gr_12.gif] where fundamental theorem of line integrals quiz _gr_13.gif] is any piecewise smooth path connecting fundamental theorem of line integrals quiz _gr_14.gif] to fundamental theorem of line integrals quiz _gr_15.gif]

(4) The gravitational force field fundamental theorem of line integrals quiz _gr_16.gif] between two particles of masses fundamental theorem of line integrals quiz _gr_17.gif] and fundamental theorem of line integrals quiz _gr_18.gif] separated by a distance fundamental theorem of line integrals quiz _gr_19.gif] is modelled by

fundamental theorem of line integrals quiz _gr_20.gif]

where fundamental theorem of line integrals quiz _gr_21.gif] and fundamental theorem of line integrals quiz _gr_22.gif] is the gravitational constant. (a) Show that fundamental theorem of line integrals quiz _gr_23.gif] is conservative by finding a scalar potential for fundamental theorem of line integrals quiz _gr_24.gif] The scalar potential function fundamental theorem of line integrals quiz _gr_25.gif] is often called the Newtonian potential. (b) Compute the amount of work done against the force field fundamental theorem of line integrals quiz _gr_26.gif] in moving an object from the point fundamental theorem of line integrals quiz _gr_27.gif] to fundamental theorem of line integrals quiz _gr_28.gif]

(5) Let

fundamental theorem of line integrals quiz _gr_29.gif]

(a) Compute the line integral fundamental theorem of line integrals quiz _gr_30.gif] where fundamental theorem of line integrals quiz _gr_31.gif] is the upper semicircle fundamental theorem of line integrals quiz _gr_32.gif] transversed counterclockwise. What is the value of fundamental theorem of line integrals quiz _gr_33.gif] if fundamental theorem of line integrals quiz _gr_34.gif] is the lower semicircle fundamental theorem of line integrals quiz _gr_35.gif] also transversed counterclockwise?

(b) Show that if fundamental theorem of line integrals quiz _gr_36.gif],  then

fundamental theorem of line integrals quiz _gr_37.gif]

but fundamental theorem of line integrals quiz _gr_38.gif] is not conservative on the unit disk fundamental theorem of line integrals quiz _gr_39.gif]

(6) An object of mass fundamental theorem of line integrals quiz _gr_40.gif] moves along a trajectory fundamental theorem of line integrals quiz _gr_41.gif] with velocity fundamental theorem of line integrals quiz _gr_42.gif] in a conservative vector field fundamental theorem of line integrals quiz _gr_43.gif] Let fundamental theorem of line integrals quiz _gr_44.gif] and fundamental theorem of line integrals quiz _gr_45.gif] be the initial and terminal points on the trajectory. (a) Show that the work done on the object is fundamental theorem of line integrals quiz _gr_46.gif] where fundamental theorem of line integrals quiz _gr_47.gif] is the object's kinetic energy. (b) Let fundamental theorem of line integrals quiz _gr_48.gif] be a scalar potential for fundamental theorem of line integrals quiz _gr_49.gif] Then fundamental theorem of line integrals quiz _gr_50.gif] is the potential energy of the object. Prove the law of conservation of energy,  namely; fundamental theorem of line integrals quiz _gr_51.gif]

Cite this as:
Fundamental Theorem Of Line Integrals Quiz
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
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