Explain why or why not Determine whether the following statementsare true and give an explanation or counterexample.a. If F = ⟨ -y, x⟩ and C is the circle of radius 4 centered at(1, 0) oriented counterclockwise, then ∮C F ⋅ dr = 0.b. If F = ⟨x, -y⟩ and C is the circle of radius 4 centered at(1, 0) oriented counterclockwise, then ∮C F ⋅ dr = 0.c. A constant vector field is conservative on ℝ2.d. The vector field F = ⟨ƒ(x), g(y)⟩ is conservative on ℝ2(assume ƒ and g are defined for all real numbers).e. Gradient fields are conservative.
Ratios
A ratio is a comparison between two numbers of the same kind. It represents how many times one number contains another. It also represents how small or large one number is compared to the other.
Trigonometric Ratios
Trigonometric ratios give values of trigonometric functions. It always deals with triangles that have one angle measuring 90 degrees. These triangles are right-angled. We take the ratio of sides of these triangles.
Explain why or why not Determine whether the following statements
are true and give an explanation or counterexample.
a. If F = ⟨ -y, x⟩ and C is the circle of radius 4 centered at
(1, 0) oriented counterclockwise, then ∮C F ⋅ dr = 0.
b. If F = ⟨x, -y⟩ and C is the circle of radius 4 centered at
(1, 0) oriented counterclockwise, then ∮C F ⋅ dr = 0.
c. A constant
d. The vector field F = ⟨ƒ(x), g(y)⟩ is conservative on ℝ2
(assume ƒ and g are defined for all real numbers).
e. Gradient fields are conservative.
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