Concep Review Ch. 16 Flashcards
What is a vector field? Give Examples with real meaning
A vector field is a function that assigns a vector to each point in its domain.
A vector field can represent the wind velocity at any location in a space, the speed an direction of ocean current at a space, or the force vector of Earth’s gravitational fields.
What is a conservative Field?
A conservative vector Field F is a vector field that is the gradient of some scalar function f, that is F = ▼f.
What is a potential function?
The function f in F = ▼f is called a potential function for F.
How do you evaluate a line integral?
The definition of a line integral is written the same way as we wrote the normal integrals with Riemann sums.
Evaluate a Line Integral a = curve by:
∫ₐ f(x,y,z) ds = ∫ₐ f (x(t), y(t), z(t) ) √ (dx/dt² + dy/dt² + dz/dt² ) dt.
Remember like in the homework if you have dx, then you can just cancel out dx.
Write Expressions for the mass and center of mass of a thin wire shaped like curve C if the wire has a linear density p(x,y)
m = ∫ₐ p(x,y) ds
X = 1/m ∫ₐ x * p(x,y) ds
Y = 1/m ∫ₐ y * p(x,y) ds
Center of Mass = (X, Y).
How to evaluate line integrals along C of a scalar function f with respect to x, y, and z?
∫ₐ f(x, y, z) dx = ∫ₐ f( x(t), y(t), z(t)) x’(t) dt
∫ₐ f(x, y, z) dy = ∫ₐ f( x(t), y(t), z(t)) y’(t) dt
∫ₐ f(x, y, z) dz = ∫ₐ f( x(t), y(t), z(t)) z’(t) dt
Define the line integral of a vector Field F along a smooth curve C given by a vector function r(t).
If F is a continuous vector field and C is given by a vector function r(t), a <= t <= b, then
∫ₐ F • dr = ∫ₐ F(r(t)) • r’(t) dt = ∫ₐ F • T
If F is a force field, then what does the line integral represent?
The work done F in moving a particle along the curve C.
Fundamental Theorem for Line Integrals
If ‘a’ is a smooth curve given by r(t), a <= t <= b and f is a differentiable function whose gradient vector ▼f. is continuous on ‘a’, then
∫ₐ ▼f • dr = f(r(b)) - f(r(a))
What does it mean for ∫ₐ F • dr to be independent of path?
∫ₐ F • dr is independent of path if the line integral has the same value for any two curves that have the same initial points and the same terminal points.
If you know that ∫ₐ F • dr is independent of path, then what can you say about F?
F is a conservative vector field, there exists a function f such that ▼f = f.
