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Fundamental Theorem of Line Integrals

In this topic:

    (1) State the fundamental theorem of line integrals.
    
    (2) Find the work done by the gravitational field (Newton's law of gravitation)   

fundamental theorem of line integrals _gr_1.gif]

where fundamental theorem of line integrals _gr_2.gif] and fundamental theorem of line integrals _gr_3.gif] are the masses of two objects and fundamental theorem of line integrals _gr_4.gif] is the gravitational constant, in moving a particle from the point fundamental theorem of line integrals _gr_5.gif] to the point fundamental theorem of line integrals _gr_6.gif] along a piecewise smooth curve fundamental theorem of line integrals _gr_7.gif]
    

Proposition (Fundamental Theorem of Line Integrals) Let fundamental theorem of line integrals _gr_8.gif] be a piecewise smooth curve that is parametrized by the vector function fundamental theorem of line integrals _gr_9.gif] for fundamental theorem of line integrals _gr_10.gif] and let fundamental theorem of line integrals _gr_11.gif] be a vector field that is continuous on fundamental theorem of line integrals _gr_12.gif] If fundamental theorem of line integrals _gr_13.gif] is a scalar function such that fundamental theorem of line integrals _gr_14.gif] then   

fundamental theorem of line integrals _gr_15.gif]

where fundamental theorem of line integrals _gr_16.gif] and fundamental theorem of line integrals _gr_17.gif] are the endpoints of fundamental theorem of line integrals _gr_18.gif]

    Proof. Suppose fundamental theorem of line integrals _gr_19.gif] and let fundamental theorem of line integrals _gr_20.gif] be the composite function fundamental theorem of line integrals _gr_21.gif] We have,

fundamental theorem of line integrals _gr_22.gif]

fundamental theorem of line integrals _gr_23.gif]

fundamental theorem of line integrals _gr_24.gif]

fundamental theorem of line integrals _gr_25.gif]

fundamental theorem of line integrals _gr_26.gif]

fundamental theorem of line integrals _gr_27.gif]

fundamental theorem of line integrals _gr_28.gif]
fundamental theorem of line integrals _gr_29.gif]

Example (Fundamental Theorem of Line Integrals) Find the work done by the gravitational field (Newton's law of gravitation)   

fundamental theorem of line integrals _gr_30.gif]

where fundamental theorem of line integrals _gr_31.gif] and fundamental theorem of line integrals _gr_32.gif] are the masses of two objects and fundamental theorem of line integrals _gr_33.gif] is the gravitational constant, in moving a particle from the point fundamental theorem of line integrals _gr_34.gif] to the point fundamental theorem of line integrals _gr_35.gif] along a piecewise smooth curve fundamental theorem of line integrals _gr_36.gif]

    Solution. We have fundamental theorem of line integrals _gr_37.gif] where    fundamental theorem of line integrals _gr_38.gif] Therefore,   

fundamental theorem of line integrals _gr_39.gif]
    
fundamental theorem of line integrals _gr_40.gif]
    
fundamental theorem of line integrals _gr_41.gif]
fundamental theorem of line integrals _gr_42.gif]

Cite this as:
Fundamental Theorem Of Line Integrals
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/fundamental-theorem-of-line-integrals.html
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