If the vector field in Example 1c describes the velocity of a fluid and place a small cork in the plane at (2, 0), what path will it follow? Example 1 Vector fields Sketch representative vectors of the following vector fields. a. F( x, y ) = 〈 0 , x 〉 = x j (a shear field) b. F( x, y ) = 〈 1 − y 2 , 0 〉 = (1 − y 2 ) i , for | y | ≤ 1(channel flow) c. F( x, y ) = 〈 − y , x 〉 = − y i + x j (a rotation field)
If the vector field in Example 1c describes the velocity of a fluid and place a small cork in the plane at (2, 0), what path will it follow? Example 1 Vector fields Sketch representative vectors of the following vector fields. a. F( x, y ) = 〈 0 , x 〉 = x j (a shear field) b. F( x, y ) = 〈 1 − y 2 , 0 〉 = (1 − y 2 ) i , for | y | ≤ 1(channel flow) c. F( x, y ) = 〈 − y , x 〉 = − y i + x j (a rotation field)
If the vector field in Example 1c describes the velocity of a fluid and place a small cork in the plane at (2, 0), what path will it follow?
Example 1 Vector fields
Sketch representative vectors of the following vector fields.
a. F(x, y) =
〈
0
,
x
〉
= xj (a shear field)
b. F(x, y) =
〈
1
−
y
2
,
0
〉
= (1 − y2)i, for |y| ≤ 1(channel flow)
c. F(x, y) =
〈
−
y
,
x
〉
= −yi + xj (a rotation field)
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.
Format:
Find the directional derivative of f(x, y, z) = -4x²z - 11 xy + 2z² at P = (-3, -3, -2) in the direction of (6,-2,0).
D(6,-2,0)| (-3,-3,-2)
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ab
sin (a)
Əə
dx
What is the maximum rate of increase of f at P?
|Vƒl(-3,-3,-2)
sin (a)
Ә
2
f
W
∞
f
8
a Ω
a
Ω
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191AfA, B are two differentiable vector point functions then prove that
curl(A × B) = A (div B) – B (div A) + (B.V) A - (A V) B,
%3D
Let x be an n-vector. (a) Is it possible for x · x to be negative? Explain. (b) If x · x = 0, what is x?
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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