(a) Define the line integral of a vector field F along a smooth curve C given by a vector function r(t).
(b) If F is a force field, what does this line integral represent?
(c) If F = ⟨P, Q, R⟩, what is the connection between the line integral of F and the line integrals of the component functions P, Q, and R?
To define: The line integral of vector field along smooth curve C.
The integrals which are done over a curve instead of a interval are referred as line integrals.
Consider a smooth curve C, given by vector function with parametric equations,
The first derivative of vector function is continuous, and not equal to zero, as curve C is a smooth curve. Divide the parametric interval of curve C which is , into the equal width of n subintervals as and subarcs of curve C as .
Hence, the parametric equations at simple point for ith subarc are,
Consider a vector field which includes the domain of curve C. Then the line integral of function is equal to the sum of product of value of f at point and subarc length .
The line integral of vector field over smooth curve C is,
Consider the length of subarc as L.
Re-modify the equation (1).
Write the expression for length of C on .
Substitute for L in equation (2),
Write the expression for .
is a unit tangent vector at point .
Write the expression for unit tangent vector .
is magnitude of first derivative of vector function
To explain: If the vector field is force, then what does the line integral represent.
To explain: The relation between the line integral of and line integrals of component functions P, Q, and R.
Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!Get Started
Calculus (MindTap Course List)
Finite Mathematics and Applied Calculus (MindTap Course List)
Calculus: An Applied Approach (MindTap Course List)
Calculus: Early Transcendental Functions
Single Variable Calculus: Early Transcendentals, Volume I
Precalculus: Mathematics for Calculus (Standalone Book)
Essentials Of Statistics
Essentials of Statistics for The Behavioral Sciences (MindTap Course List)
Calculus (MindTap Course List)
Statistics for The Behavioral Sciences (MindTap Course List)
Single Variable Calculus
Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach
Probability and Statistics for Engineering and the Sciences
Understanding Basic Statistics
Mathematical Applications for the Management, Life, and Social Sciences
Elementary Technical Mathematics
Elementary Geometry For College Students, 7e
Mathematical Excursions (MindTap Course List)
Contemporary Mathematics for Business & Consumers
Single Variable Calculus: Early Transcendentals
Calculus of a Single Variable
Calculus: Early Transcendental Functions (MindTap Course List)
Trigonometry (MindTap Course List)
Elements Of Modern Algebra
Study Guide for Stewart's Multivariable Calculus, 8th
Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th
Finite Mathematics for the Managerial, Life, and Social Sciences
Elementary Geometry for College Students