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.
University Calculus: Early Transcendentals (4th Edition)
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