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4. Match the vector fields given below with their plots. (a) F(x,y, z) =< x, Y, z > (b) F(x, y, z) =< 0, 1,0 > (c) F(x, y, 2) = I² + y²' VI² + y² - (d) F(x, y, z) = (x² + y² + z²)3/2' (x² + y² + z²)3/2' (2² + y² + z?)3/2, Field (I) Field (II) Field (III) Field (VI)

4. Match the vector fields given below with their plots. (a) F(x,y, z) =< x, Y, z > (b) F(x, y, z) =< 0, 1,0 > (c) F(x, y, 2) = I² + y²' VI² + y² - (d) F(x, y, z) = (x² + y² + z²)3/2' (x² + y² + z²)3/2' (2² + y² + z?)3/2, Field (I) Field (II) Field (III) Field (VI)

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Vector Field Matching** The task is to match the given vector fields with their respective plots. ### Vector Fields: (a) \(\vec{F}(x, y, z) = \langle x, y, z \rangle\) (b) \(\vec{F}(x, y, z) = \langle 0, 1, 0 \rangle\) (c) \(\vec{F}(x, y, z) = \left\langle \frac{y}{\sqrt{x^2 + y^2}}, \frac{-x}{\sqrt{x^2 + y^2}}, 0 \right\rangle\) (d) \(\vec{F}(x, y, z) = \left\langle \frac{x}{(x^2+y^2+z^2)^{3/2}}, \frac{y}{(x^2+y^2+z^2)^{3/2}}, \frac{z}{(x^2+y^2+z^2)^{3/2}} \right\rangle\) ### Plots: - **Field (I):** - The plot shows vectors radiating outward from the origin. The length of the vectors increases with distance from the origin, representing the vector field \(\vec{F}(x, y, z) = \langle x, y, z \rangle\). - **Field (II):** - This plot illustrates vectors that are directed in a uniform horizontal direction. The vectors are parallel, and their magnitude remains constant, which matches the vector field \(\vec{F}(x, y, z) = \langle 0, 1, 0 \rangle\). - **Field (III):** - This plot displays a swirling pattern around the z-axis, with vectors forming circular paths in the xy-plane. This corresponds to the vector field \(\vec{F}(x, y, z) = \left\langle \frac{y}{\sqrt{x^2 + y^2}}, \frac{-x}{\sqrt{x^2 + y^2}}, 0 \right\rangle\). - **Field (VI):** - The plot shows vectors directed towards the origin, decreasing in magnitude with distance from the origin. This matches the vector field \(\vec{F}(x, y, z)
Transcribed Image Text:**Vector Field Matching** The task is to match the given vector fields with their respective plots. ### Vector Fields: (a) \(\vec{F}(x, y, z) = \langle x, y, z \rangle\) (b) \(\vec{F}(x, y, z) = \langle 0, 1, 0 \rangle\) (c) \(\vec{F}(x, y, z) = \left\langle \frac{y}{\sqrt{x^2 + y^2}}, \frac{-x}{\sqrt{x^2 + y^2}}, 0 \right\rangle\) (d) \(\vec{F}(x, y, z) = \left\langle \frac{x}{(x^2+y^2+z^2)^{3/2}}, \frac{y}{(x^2+y^2+z^2)^{3/2}}, \frac{z}{(x^2+y^2+z^2)^{3/2}} \right\rangle\) ### Plots: - **Field (I):** - The plot shows vectors radiating outward from the origin. The length of the vectors increases with distance from the origin, representing the vector field \(\vec{F}(x, y, z) = \langle x, y, z \rangle\). - **Field (II):** - This plot illustrates vectors that are directed in a uniform horizontal direction. The vectors are parallel, and their magnitude remains constant, which matches the vector field \(\vec{F}(x, y, z) = \langle 0, 1, 0 \rangle\). - **Field (III):** - This plot displays a swirling pattern around the z-axis, with vectors forming circular paths in the xy-plane. This corresponds to the vector field \(\vec{F}(x, y, z) = \left\langle \frac{y}{\sqrt{x^2 + y^2}}, \frac{-x}{\sqrt{x^2 + y^2}}, 0 \right\rangle\). - **Field (VI):** - The plot shows vectors directed towards the origin, decreasing in magnitude with distance from the origin. This matches the vector field \(\vec{F}(x, y, z)
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