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Match the vector fields given below with their plots. (a) F(x, y) = 2yi + 2x3 (c) F(x, y) = (sin y)i + (sin x)j (b) F(x, y) = (x + y)i+ (x – y)3. (d) F(x, y) = rỉ + y°j. 27 %3D -1 -4. - 2 -6 -4 -2 0 2 -1 2 2 2 1 -1 -1 -2 -2 -2 1 2 -2 1 2.

Match the vector fields given below with their plots. (a) F(x, y) = 2yi + 2x3 (c) F(x, y) = (sin y)i + (sin x)j (b) F(x, y) = (x + y)i+ (x – y)3. (d) F(x, y) = rỉ + y°j. 27 %3D -1 -4. - 2 -6 -4 -2 0 2 -1 2 2 2 1 -1 -1 -2 -2 -2 1 2 -2 1 2.

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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**Exercise 2: Matching Vector Fields to Plots** Match the given vector fields with their corresponding plots. **Vector Fields:** (a) \(\vec{F}(x, y) = 2y\hat{i} + 2x\hat{j}\) (b) \(\vec{F}(x, y) = (x + y)\hat{i} + (x - y)\hat{j}\) (c) \(\vec{F}(x, y) = (\sin y)\hat{i} + (\sin x)\hat{j}\) (d) \(\vec{F}(x, y) = x\hat{i} + y^2\hat{j}\) **Plots Explanation:** - **Top Left Plot:** The vectors tend to point outward from the center, indicating a radial pattern with symmetry around both axes. The vector magnitudes increase as they move away from the origin. - **Top Right Plot:** This plot features vectors swirling in a pattern, suggesting rotation or circulation behavior, with differing directions depending on the quadrant. - **Bottom Left Plot:** The vectors show a wave-like pattern both horizontally and vertically, suggesting a sinusoidal behavior, especially reflecting changes based on positions along each axis. - **Bottom Right Plot:** A distinct pattern shows vectors pointing in directions that are influenced primarily by their position's x-component but with some vertical asymmetry due to the involvement of y. The task is to analyze each vector field equation to identify which plot it corresponds to based on the directional and magnitude characteristics depicted in the plots.
Transcribed Image Text:**Exercise 2: Matching Vector Fields to Plots** Match the given vector fields with their corresponding plots. **Vector Fields:** (a) \(\vec{F}(x, y) = 2y\hat{i} + 2x\hat{j}\) (b) \(\vec{F}(x, y) = (x + y)\hat{i} + (x - y)\hat{j}\) (c) \(\vec{F}(x, y) = (\sin y)\hat{i} + (\sin x)\hat{j}\) (d) \(\vec{F}(x, y) = x\hat{i} + y^2\hat{j}\) **Plots Explanation:** - **Top Left Plot:** The vectors tend to point outward from the center, indicating a radial pattern with symmetry around both axes. The vector magnitudes increase as they move away from the origin. - **Top Right Plot:** This plot features vectors swirling in a pattern, suggesting rotation or circulation behavior, with differing directions depending on the quadrant. - **Bottom Left Plot:** The vectors show a wave-like pattern both horizontally and vertically, suggesting a sinusoidal behavior, especially reflecting changes based on positions along each axis. - **Bottom Right Plot:** A distinct pattern shows vectors pointing in directions that are influenced primarily by their position's x-component but with some vertical asymmetry due to the involvement of y. The task is to analyze each vector field equation to identify which plot it corresponds to based on the directional and magnitude characteristics depicted in the plots.
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