Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. For a function f of a single variable, if f ′( x ) = 0 for all x in the domain, then f is a constant function. If ▿ ·F = 0 for all points in the domain, then F is constant. b. If ▿ × F = 0 , then F is constant. c. A vector field consisting of parallel vectors has zero curl. d. A vector field consisting of parallel vectors has zero divergence. e. curl F is orthogonal to F .
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. For a function f of a single variable, if f ′( x ) = 0 for all x in the domain, then f is a constant function. If ▿ ·F = 0 for all points in the domain, then F is constant. b. If ▿ × F = 0 , then F is constant. c. A vector field consisting of parallel vectors has zero curl. d. A vector field consisting of parallel vectors has zero divergence. e. curl F is orthogonal to F .
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. For a function f of a single variable, if f′(x) = 0 for all x in the domain, then f is a constant function. If ▿ ·F = 0 for all points in the domain, then F is constant.
b. If ▿ × F = 0, then F is constant.
c. A vector field consisting of parallel vectors has zero curl.
d. A vector field consisting of parallel vectors has zero divergence.
e. curl F is orthogonal to F.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
Q.1. The set of all positive real numbers with the operations
x + y = xy
kx = xk
Is it a vector space? Justify your answer.
Consider two vector fields A and B. The vector field B is solenoidal. Use subscript notation
to simplify
(A × V) × B – A x curl B.
You may assume the relation ɛijkƐ klm
= 8;18jm – dim8ji.
You are on a rollercoaster, and the path of your body is modeled by a vector function r(t),
where t is in seconds, the units of distance are in feet, and t = 0 represents the start of the
ride. Assume the axes represent the standard cardinal directions and elevation (x is E/W, y
is N/S, z is height). Explain what the following would represent physically, being as specific
as possible. These are all common roller coaster shapes/behaviors and can be explained in
specific language with regard to units:
a. r(0)=r(120)
b. For 0 ≤ t ≤ 30, N(t) = 0
c. r'(30) = 120
d. For 60 ≤ t ≤ 64, k(t) =
40
and z is constant.
e.
For 100 ≤ t ≤ 102, your B begins by pointing toward positive z, and does one full
rotation in the normal (NB) plane while your T remains constant.
Chapter 17 Solutions
Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)
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