Vector fields in polar coordinates A vector field in polar coordinates has the form F ( r , θ ) = F ( r , θ ) u r + g ( r , θ ) u θ , where the unit vectors are defined in Exercise 62 . Sketch the following vector fields and express them in Cartesian coordinates. 62. Vectors in ℝ 2 may also be expressed in terms of polar coordinates. The standard coordinate unit vectors in polar coordinates are denoted u r and u θ (see figure). Unlike the coordinate unite vectors in Cartesian coordinates, u r and u θ change their direction depending on the point ( r , θ ). Use the figure to show that for r > 0, the following relationships among the unit vectors in Cartesian and polar coordinates hold: u r = cos θ i + sin θ j i = u r cos θ − u θ sin θ u θ = sin θ i + cos θ j j = u r sin θ + u θ cos θ 66. F = r u θ
Vector fields in polar coordinates A vector field in polar coordinates has the form F ( r , θ ) = F ( r , θ ) u r + g ( r , θ ) u θ , where the unit vectors are defined in Exercise 62 . Sketch the following vector fields and express them in Cartesian coordinates. 62. Vectors in ℝ 2 may also be expressed in terms of polar coordinates. The standard coordinate unit vectors in polar coordinates are denoted u r and u θ (see figure). Unlike the coordinate unite vectors in Cartesian coordinates, u r and u θ change their direction depending on the point ( r , θ ). Use the figure to show that for r > 0, the following relationships among the unit vectors in Cartesian and polar coordinates hold: u r = cos θ i + sin θ j i = u r cos θ − u θ sin θ u θ = sin θ i + cos θ j j = u r sin θ + u θ cos θ 66. F = r u θ
Solution Summary: The author explains how to sketch the vector field and express it in Cartesian coordinates.
Vector fields in polar coordinates A vector field in polar coordinates has the form F(r, θ) = F(r, θ) ur + g(r, θ) uθ, where the unit vectors are defined in Exercise 62. Sketch the following vector fields and express them in Cartesian coordinates.
62. Vectors in ℝ2 may also be expressed in terms of polar coordinates. The standard coordinate unit vectors in polar coordinates are denoted ur and uθ (see figure). Unlike the coordinate unite vectors in Cartesian coordinates, ur and uθ change their direction depending on the point (r, θ). Use the figure to show that for r > 0, the following relationships among the unit vectors in Cartesian and polar coordinates hold:
ur = cos θi + sin θj i = ur cos θ − uθ sin θ
uθ = sin θi + cos θj j = ur sin θ + uθ cos θ
66. F = ruθ
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.
2. Vector Addition and Vector Components: Vector A is 2.80 cm long and is 60° above the x-axis in the
first quadrant. Vector B is 1.90 cm long and is 60° below the x-axis in the fourth quadrant. Use components
to find the magnitude and direction of (a) Ā + B; (b) Ā -B (c) B+ A. In each case, (d) sketch the vector
addition or subtraction and show that your numerical answers are in qualitative agreement with your
sketch.
1. (a) A vector in the plane is a line segment with an assigned
direction. In Figure I below, the vector u has initial point
. and terminal point .
vectors 2u and u + v.
Sketch the
(b) A vector in a coordinate plane is expressed by using
components. In Figure II below, the vector u has initial
point (,D and terminal point (.). In compo-
nent form we write u = (.), and v =
Then 2u = (. ) and u + v = (
D
B
u
I
II
University Calculus: Early Transcendentals (4th Edition)
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