Fall 2019Math 201: Calculus and Analytic Geometry III
Lecture 8: Sections 26-28
Lecture Overview
This lecture covers key concepts from Calculus III including vector fields, line integrals, and surface integrals. We'll explore these topics with a focus on practical applications and problem-solving techniques.
Section 26: Vector Fields
Definition
A vector field is a function that assigns a vector to each point in space. In two dimensions, we can represent a vector field as F(x, y) = M(x, y)i + N(x, y)j, where M and N are functions of x and y.
Properties of Conservative Vector Fields
A vector field F is conservative if:
- It is the gradient of some scalar function f (F = f)
- It has path-independent line integrals
- Its curl is zero ( F = 0)
- For simply connected domains, these conditions are equivalent
Example 26.1
Determine whether the vector field F(x, y) = (2xy + y)i + (x + 2xy)j is conservative.
Solution: We can check if M/y = N/x.
M/y = /y(2xy + y) = 2x + 2y
N/x = /x(x + 2xy) = 2x + 2y
Since M/y = N/x, the vector field is conservative.
Applications
Vector fields have numerous applications in physics, including:
- Electromagnetic fields
- Fluid dynamics
- Gravitational fields
- Heat transfer
Section 27: Line Integrals
Definition
A line integral is an integral where the function to be integrated is evaluated along a curve. There are several types of line integrals:
- Scalar line integrals: C f(x, y, z) ds
- Vector line integrals: C F dr
For a curve parameterized by r(t) = (x(t), y(t), z(t)) for a t b, the line integral becomes:
C F dr = [a to b] F(r(t)) r'(t) dt
Fundamental Theorem for Line Integrals
For a conservative vector field F = f, we have:
C f dr = f(r(b)) - f(r(a))
Example 27.1
Calculate C yx dx + xy dy along the path from (0, 0) to (1, 1) given by y = x.
Solution: Parameterize the curve as x = t, y = t for 0 t 1.
Then dx = dt and dy = 2t dt.
Substituting into the integral:
[0 to 1] (t)tdt + t(t)2tdt
= [0 to 1] (t + 2t) dt
= [t/6 + 2t/7] from 0 to 1
= 1/6 + 2/7 - 0 = 19/42
Applications
Line integrals are used to calculate:
- Work done by a force field
- Mass of a wire with variable density
- Circulation of fluids
Electromagnetic work
Section 28: Surface Integrals
Definition
Surface integrals extend line integrals to two-dimensional surfaces in three-dimensional space. A surface integral of a scalar function f over a surface S is given by:
S f(x, y, z) dS = D f(r(u, v)) |r_u r_v| dA
where D is the parameter domain and r(u, v) is a parameterization of the surface.
Types of Surface Integrals
- Scalar surface integrals: S f(x, y, z) dS
- Vector surface integrals (flux): S F n dS
Calculating Flux Through a Surface
The flux of a vector field F across a surface S with unit normal vector n is:
Flux = S F n dS
Example 28.1
Calculate the flux of F = xi + yj + zk through the upper hemisphere x + y + z = 4, z 0.
Solution: Use spherical coordinates to parameterize the surface:
x = 2sin()cos(), y = 2sin()sin(), z = 2cos()
For 0 2, 0 /2.
The outward unit normal is n = (x/2)i + (y/2)j + (z/2)k = sin()cos()i + sin()sin()j + cos()k.
F n = xsin()cos() + ysin()sin() + zcos()
= 4sin()cos()sin()cos() + 4sin()sin()sin()sin() + 4cos()cos()
= 4sin()cos() + 4sin()sin() + 4cos()
= 4sin() + 4cos() = 4
Flux = S F n dS = [0 to 2][0 to /2] 42sin() d d
= 16[0 to 2][0 to /2] sin() d d
= 16[0 to 2] [-cos()][0 to /2] d
= 16[0 to 2] [0 - (-1)] d
= 16[0 to 2] 1 d = 32
Applications
Surface integrals are used to calculate:
- Mass of a surface with variable density
- Fluid flow across a surface
- Electromagnetic flux
- Heat flow across a boundary
Important Theorems
Green's Theorem
D F dr = D (N/x - M/y) dA
where F = Mi + Nj is a vector field on an open region containing D, and D is the positively oriented boundary curve of D.
Stokes' Theorem
S F dr = S ( F) n dS
relating the line integral of a vector field around a boundary curve S to the surface integral of the curl of the vector field over the surface S.
Divergence Theorem
E F n dS = E ( F) dV
relating the flux of a vector field through a closed surface E to the divergence of the field within the enclosed volume E.
Practice Problems
Problem 1:
Determine whether the vector field F(x, y) = (y - 2x)i + (2xy + 3)j is conservative. If it is, find a potential function.
Problem 2:
Calculate C 2xy dx + xy dy, where C is the triangle with vertices (0, 0), (1, 0), and (1, 2).
Problem 3:
Find the flux of F = xi - j + zk through the surface S: x + 2y + 3z = 6 in the first octant, in the direction away from the origin.
Lecture Summary
In Lecture 8, we explored vector fields, line integrals, and surface integrals - fundamental concepts in multivariable calculus. We learned how to determine if a vector field is conservative and how to calculate various types of integrals over curves and surfaces. These tools provide powerful methods for solving physical problems involving work, circulation, flux, and other important quantities. The theorems of Green, Stokes, and the Divergence Theorem connect seemingly different types of integrals and are essential for advanced applications in mathematics, physics, and engineering.
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