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The Fundamental Theorem and Path Independence Homework
Directions: Write legibly and in pencil. Complete the homework on time and by yourself. For each problem, write the instructions, label the solution, show all steps, and write the final answer in a sentence. Do not turn in your scratch work. Staple your pages together, in the correct order, and use this page as a cover sheet.
(1) Determine whether or not the vector field
![the fundamental theorem and path independence homework _gr_1.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_1.gif){kind=small}
is conservative.
(2) Determine whether or not the vector field
![the fundamental theorem and path independence homework _gr_2.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_2.gif){kind=small}
is conservative.
(3) Determine whether or not the vector field
![the fundamental theorem and path independence homework _gr_3.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_3.gif){kind=small}
is conservative.
(4) Determine whether or not the vector field
![the fundamental theorem and path independence homework _gr_4.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_4.gif){kind=small}
is conservative.
(5) Determine whether or not the vector field
![the fundamental theorem and path independence homework _gr_5.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_5.gif){kind=small}
is conservative.
(6) Determine whether or not the vector field
![the fundamental theorem and path independence homework _gr_6.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_6.gif){kind=small}
is conservative.
(7) Show that the vector field
![the fundamental theorem and path independence homework _gr_7.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_7.gif){kind=small}
is conservative and find a scalar potential function
![the fundamental theorem and path independence homework _gr_8.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_8.gif){kind=small}
for
![the fundamental theorem and path independence homework _gr_9.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_9.gif){kind=small}
Then evaluate the line integral
![the fundamental theorem and path independence homework _gr_10.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_10.gif){kind=small}
where
![the fundamental theorem and path independence homework _gr_11.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_11.gif){kind=small}
is any smooth path connecting
![the fundamental theorem and path independence homework _gr_12.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_12.gif){kind=small}
to
![the fundamental theorem and path independence homework _gr_13.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_13.gif){kind=small}
(8) Show that the vector field
![the fundamental theorem and path independence homework _gr_14.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_14.gif){kind=small}
is conservative and find a scalar potential function
![the fundamental theorem and path independence homework _gr_15.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_15.gif){kind=small}
for
![the fundamental theorem and path independence homework _gr_16.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_16.gif){kind=small}
Then evaluate the line integral
![the fundamental theorem and path independence homework _gr_17.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_17.gif){kind=small}
where
![the fundamental theorem and path independence homework _gr_18.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_18.gif){kind=small}
is any smooth path connecting
![the fundamental theorem and path independence homework _gr_19.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_19.gif){kind=small}
to
![the fundamental theorem and path independence homework _gr_20.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_20.gif){kind=small}
(9) Show that the vector field
![the fundamental theorem and path independence homework _gr_21.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_21.gif){kind=small}
is conservative and find a scalar potential function
![the fundamental theorem and path independence homework _gr_22.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_22.gif){kind=small}
for
![the fundamental theorem and path independence homework _gr_23.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_23.gif){kind=small}
Then evaluate the line integral
![the fundamental theorem and path independence homework _gr_24.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_24.gif){kind=small}
where
![the fundamental theorem and path independence homework _gr_25.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_25.gif){kind=small}
is any smooth path connecting
![the fundamental theorem and path independence homework _gr_26.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_26.gif){kind=small}
to
![the fundamental theorem and path independence homework _gr_27.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_27.gif){kind=small}
(10) Show that the vector field
![the fundamental theorem and path independence homework _gr_28.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_28.gif){kind=small}
is conservative and find a scalar potential function
![the fundamental theorem and path independence homework _gr_29.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_29.gif){kind=small}
for
![the fundamental theorem and path independence homework _gr_30.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_30.gif){kind=small}
(11) Show that the vector field
![the fundamental theorem and path independence homework _gr_31.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_31.gif){kind=small}
is conservative and find a scalar potential function
![the fundamental theorem and path independence homework _gr_32.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_32.gif){kind=small}
for
![the fundamental theorem and path independence homework _gr_33.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_33.gif){kind=small}
(12) Show that the vector field
![the fundamental theorem and path independence homework _gr_34.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_34.gif){kind=small}
is conservative and find a scalar potential function
![the fundamental theorem and path independence homework _gr_35.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_35.gif){kind=small}
for
![the fundamental theorem and path independence homework _gr_36.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_36.gif){kind=small}
Then evaluate the line integral
![the fundamental theorem and path independence homework _gr_37.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_37.gif){kind=small}
where
![the fundamental theorem and path independence homework _gr_38.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_38.gif){kind=small}
is any smooth path connecting
![the fundamental theorem and path independence homework _gr_39.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_39.gif){kind=small}
to
![the fundamental theorem and path independence homework _gr_40.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_40.gif){kind=small}
(13) Show that the vector field
![the fundamental theorem and path independence homework _gr_41.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_41.gif){kind=small}
is conservative and find a scalar potential
![the fundamental theorem and path independence homework _gr_42.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_42.gif){kind=small}
for
![the fundamental theorem and path independence homework _gr_43.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_43.gif){kind=small}
Then evaluate the line integral
![the fundamental theorem and path independence homework _gr_44.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_44.gif){kind=small}
where
![the fundamental theorem and path independence homework _gr_45.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_45.gif){kind=small}
is any smooth path connecting
![the fundamental theorem and path independence homework _gr_46.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_46.gif){kind=small}
to
![the fundamental theorem and path independence homework _gr_47.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_47.gif){kind=small}
(14) Show that the vector field
![the fundamental theorem and path independence homework _gr_48.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_48.gif){kind=small}
is conservative and find a scalar potential
![the fundamental theorem and path independence homework _gr_49.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_49.gif){kind=small}
for
![the fundamental theorem and path independence homework _gr_50.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_50.gif){kind=small}
(15) Show that the vector field
![the fundamental theorem and path independence homework _gr_51.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_51.gif){kind=small}
is conservative and then evaluate the line integral
![the fundamental theorem and path independence homework _gr_52.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_52.gif){kind=small}
where
![the fundamental theorem and path independence homework _gr_53.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_53.gif){kind=small}
is any piecewise smooth path connecting
![the fundamental theorem and path independence homework _gr_54.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_54.gif){kind=small}
to
![the fundamental theorem and path independence homework _gr_55.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_55.gif){kind=small}
(16) The gravitational force field
![the fundamental theorem and path independence homework _gr_56.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_56.gif){kind=small}
between two particles of masses
![the fundamental theorem and path independence homework _gr_57.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_57.gif){kind=small}
and
![the fundamental theorem and path independence homework _gr_58.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_58.gif){kind=small}
separated by a distance
![the fundamental theorem and path independence homework _gr_59.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_59.gif){kind=small}
is modelled by
![the fundamental theorem and path independence homework _gr_60.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_60.gif){kind=small}
where
![the fundamental theorem and path independence homework _gr_61.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_61.gif){kind=small}
and
![the fundamental theorem and path independence homework _gr_62.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_62.gif){kind=small}
is the gravitational constant.
(a) Show that
![the fundamental theorem and path independence homework _gr_63.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_63.gif){kind=small}
is conservative by finding a scalar potential for
![the fundamental theorem and path independence homework _gr_64.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_64.gif){kind=small}
The scalar potential function
![the fundamental theorem and path independence homework _gr_65.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_65.gif){kind=small}
is often called the Newtonian potential.
(b) Compute the amount of work done against the force field
![the fundamental theorem and path independence homework _gr_66.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_66.gif){kind=small}
in moving an object from the point
![the fundamental theorem and path independence homework _gr_67.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_67.gif){kind=small}
to
![the fundamental theorem and path independence homework _gr_68.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_68.gif){kind=small}
(17) Let
![the fundamental theorem and path independence homework _gr_69.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_69.gif){kind=small}
(a) Compute the line integral
![the fundamental theorem and path independence homework _gr_70.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_70.gif){kind=small}
where
![the fundamental theorem and path independence homework _gr_71.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_71.gif){kind=small}
is the upper semicircle
![the fundamental theorem and path independence homework _gr_72.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_72.gif){kind=small}
transversed counterclockwise.
What is the value of
![the fundamental theorem and path independence homework _gr_73.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_73.gif){kind=small}
if
![the fundamental theorem and path independence homework _gr_74.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_74.gif){kind=small}
is the lower semicircle
![the fundamental theorem and path independence homework _gr_75.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_75.gif){kind=small}
also transversed counterclockwise?
(b) Show that if
![the fundamental theorem and path independence homework _gr_76.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_76.gif){kind=small}
, then
![the fundamental theorem and path independence homework _gr_77.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_77.gif){kind=small}
but
![the fundamental theorem and path independence homework _gr_78.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_78.gif){kind=small}
is not conservative on the unit disk
![the fundamental theorem and path independence homework _gr_79.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_79.gif){kind=small}
(18) An object of mass
![the fundamental theorem and path independence homework _gr_80.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_80.gif){kind=small}
moves along a trajectory
![the fundamental theorem and path independence homework _gr_81.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_81.gif){kind=small}
with velocity
![the fundamental theorem and path independence homework _gr_82.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_82.gif){kind=small}
in a conservative vector field
![the fundamental theorem and path independence homework _gr_83.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_83.gif){kind=small}
Let
![the fundamental theorem and path independence homework _gr_84.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_84.gif){kind=small}
and
![the fundamental theorem and path independence homework _gr_85.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_85.gif){kind=small}
be the initial and terminal points on the trajectory.
(a) Show that the work done on the object is
![the fundamental theorem and path independence homework _gr_86.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_86.gif){kind=small}
where
![the fundamental theorem and path independence homework _gr_87.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_87.gif){kind=small}
is the object's kinetic energy.
(b) Let
![the fundamental theorem and path independence homework _gr_88.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_88.gif){kind=small}
be a scalar potential for
![the fundamental theorem and path independence homework _gr_89.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_89.gif){kind=small}
Then
![the fundamental theorem and path independence homework _gr_90.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_90.gif){kind=small}
is the potential energy of the object.
Prove the law of conservation of energy, namely;
![the fundamental theorem and path independence homework _gr_91.gif]](pages/the-fundamental-theorem-and-path-independence-homework/Images/the-fundamental-theorem-and-path-independence-homework_gr_91.gif){kind=small}
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The Fundamental Theorem And Path Independence Homework
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/the-fundamental-theorem-and-path-independence-homework.html
