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Let F(r, y) = (x + 1)7, and let C be the portion of the parabola y = x² that starts at (0,0) and ends at (2, 4). (a) Without doing any calculations, state whether fF•dř° is positive, neg- ative, or zero. Clearly state your answer, and explain based on the diagram to the right how you know your answer is correct.

Let F(r, y) = (x + 1)7, and let C be the portion of the parabola y = x² that starts at (0,0) and ends at (2, 4). (a) Without doing any calculations, state whether fF•dř° is positive, neg- ative, or zero. Clearly state your answer, and explain based on the diagram to the right how you know your answer is correct.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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The problem provided is as follows: Let \(\vec{F}(x, y) = (x+1) \vec{i}\), and let \(C\) be the portion of the parabola \(y = x^2\) that starts at \((0, 0)\) and ends at \((2, 4)\). (a) Without doing any calculations, state whether \(\int_{C} \vec{F} \cdot d\vec{r}\) is positive, negative, or zero. Clearly state your answer, and explain based on the diagram to the right how you know your answer is correct. **Explanation of the diagram:** The diagram shows the curve \(C\), which is part of the parabola \(y = x^2\), ranging from \((0, 0)\) to \((2, 4)\) on the xy-plane. The curve is plotted against a field of vectors (depicted as arrows) pointing upward. These vectors, \(\vec{F}(x, y) = (x+1) \vec{i}\), are aligned along the positive x-direction and increase in magnitude as x increases. Since the vectors are always horizontal (along the x-axis) and the curve progresses from left to right while increasing in both x and y, the dot product \(\vec{F} \cdot d\vec{r}\) will be positive. This is because the direction of \(\vec{F}\) and the tangent vector along the curve are in the same general direction, resulting in a positive dot product. Therefore, \(\int_{C} \vec{F} \cdot d\vec{r}\) is positive. The alignment of the vectors with the direction of movement along the curve confirms this conclusion visually.
Transcribed Image Text:The problem provided is as follows: Let \(\vec{F}(x, y) = (x+1) \vec{i}\), and let \(C\) be the portion of the parabola \(y = x^2\) that starts at \((0, 0)\) and ends at \((2, 4)\). (a) Without doing any calculations, state whether \(\int_{C} \vec{F} \cdot d\vec{r}\) is positive, negative, or zero. Clearly state your answer, and explain based on the diagram to the right how you know your answer is correct. **Explanation of the diagram:** The diagram shows the curve \(C\), which is part of the parabola \(y = x^2\), ranging from \((0, 0)\) to \((2, 4)\) on the xy-plane. The curve is plotted against a field of vectors (depicted as arrows) pointing upward. These vectors, \(\vec{F}(x, y) = (x+1) \vec{i}\), are aligned along the positive x-direction and increase in magnitude as x increases. Since the vectors are always horizontal (along the x-axis) and the curve progresses from left to right while increasing in both x and y, the dot product \(\vec{F} \cdot d\vec{r}\) will be positive. This is because the direction of \(\vec{F}\) and the tangent vector along the curve are in the same general direction, resulting in a positive dot product. Therefore, \(\int_{C} \vec{F} \cdot d\vec{r}\) is positive. The alignment of the vectors with the direction of movement along the curve confirms this conclusion visually.
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