Evaluating line integrals Evaluate the line integral ∫ C F ⋅ d r for the following vector fields F and curves C in two ways. a. By parameterizing C b. By using the Fundamental Theorem for line integrals, if possible 26. F = 〈 x , – y 〉; C is the square with vertices (±1, ±1) with counterclockwise orientation.
Evaluating line integrals Evaluate the line integral ∫ C F ⋅ d r for the following vector fields F and curves C in two ways. a. By parameterizing C b. By using the Fundamental Theorem for line integrals, if possible 26. F = 〈 x , – y 〉; C is the square with vertices (±1, ±1) with counterclockwise orientation.
Evaluating line integralsEvaluate the line integral
∫
C
F
⋅
d
r
for the following vector fieldsFand curves C in two ways.
a. By parameterizing C
b. By using the Fundamental Theorem for line integrals, if possible
26. F = 〈x, –y〉; C is the square with vertices (±1, ±1) with counterclockwise orientation.
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.
Alternative construction of potential functions in ℝ2 Assume the vector field F is conservative on ℝ2, so that the line integral ∫C F ⋅ dr is independent of path. Use the following procedure to construct a potential function w for the vector field F = ⟨ƒ, g⟩ = ⟨2x - y, -x + 2y⟩ .
Use the procedure given above to construct potential functions for thefollowing fields.
F = ⟨x, y⟩
Line integrals of vector fields in the plane Given the followingvector fields and oriented curves C, evaluate ∫C F ⋅ T ds.
F = ⟨x, y⟩ on the parabola r(t) = ⟨4t, t2⟩ , for 0 ≤ t ≤ 1
Alternative construction of potential functions in ℝ2 Assume the vector field F is conservative on ℝ2, so that the line integral ∫C F ⋅ dr is independent of path. Use the following procedure to construct a potential function w for the vector field F = ⟨ƒ, g⟩ = ⟨2x - y, -x + 2y⟩ .a. Let A be (0, 0) and let B be an arbitrary point (x, y). Define φ(x, y) to be the work required to move an object from A to B, where φ(A) = 0. Let C1 be the path from A to (x, 0) to B, and let C2 be the path from A to (0, y) to B. Draw a picture.b. Evaluate ∫C1 F ⋅ dr = ∫C1 ƒ dx + g dy and conclude thatφ(x, y) = x2 - xy + y2.c. Verify that the same potential function is obtained by evaluatingthe line integral over C2.
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