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Evaluating a Line
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Calculus (MindTap Course List)
- find curl (curl F) = V x (V X F).arrow_forwardEvaluate F · dr using the Fundamental Theorem of Line Integrals. Use a computer algebra system to verify your results. (2z + 4y) dx + (4x – 3z) dy + (2x – 3y) dz (a) C: line segment from (0, 0, 0) to (1, 1, 1) (b) C: line segment from (0, 0, 0) to (0, 0, 1) to (1, 1, 1) (c) C: line segment from (0, 0, 0) to (1, 0, 0) to (1, 1, 0) to (1, 1, 1)arrow_forwardEvaluating a Line Integral Using Green's Theorem In Exercise, use Green's Theorem to evaluate the line integral. √(√(x² - 1²) C: r = 1 + cos 8 (x² - y²) dx + 2xy dyarrow_forward
- 人工知能を使用せず、 すべてを段階的にデジタル形式で解決してください。 ありがとう SOLVE STEP BY STEP IN DIGITAL FORMAT DON'T USE CHATGPT For Exercises 1-4, use Green's Theorem to evaluate the given line integral around the curve C, traversed counterclockwise. 1. f(x² - y²) dx + 2xydy; C is the boundary of R = {(x,y): 0≤x≤ 1, 2x² ≤ y ≤ 2x) x³y dx + 2xydy; C is the boundary of R = {(x, y): 0 ≤x≤1, x² ≤ y ≤ x} $² 2ydx-3xd y; C is the circle x² + y² = 1 2. 3. 4. ·f (ex² + y²) dx + (e¹² + x³)dy; C is the boundary of the triangle with vertices (0,0), (4,0) and (0,4)arrow_forwardDisplacement d→1 is in the yz plane 62.8 o from the positive direction of the y axis, has a positive z component, and has a magnitude of 5.10 m. Displacement d→2 is in the xz plane 37.0 o from the positive direction of the x axis, has a positive z component, and has magnitude 0.900 m. What are (a) d→1⋅d→2 , (b) the x component of d→1×d→2 , (c) the y component of d→1×d→2 , (d) the z component of d→1×d→2 , and (e) the angle between d→1 and d→2 ?arrow_forwardWhat is the geometrical meaning of an integral of a vector function?arrow_forward
- QUICK CHECK 3 Let u(t) = (t,t, t) and v(t) = (1, 1, 1). Compute d (n(t) • v(t)) using Derivative dt Rule 5, and show that it agrees with the result obtained by first computing the dot product and differentiating directly. <arrow_forwardDescribe what it means for a vector-valued function r(t) to be continuous at a point.arrow_forwardUsing a Function, (a) find the gradient of the function at P, (b) find a unit normal vector to the level curve f(x, y) = c at P, (c) find the tangent line to the level curve f(x, y) = c at P, and (d) sketch the level curve, the unit normal vector, and the tangent line in the xy-plane. f(x, y) = 9x2 − 4y2, c = 65, P(3, 2)arrow_forward
- Nonuniform straight-line motion Consider the motion of an object given by the position function r(t) = ƒ(t)⟨a, b, c⟩ + ⟨x0, y0, z0⟩, for t ≥ 0,where a, b, c, x0, y0, and z0 are constants, and ƒ is a differentiable scalar function, for t ≥ 0.a. Explain why r describes motion along a line.b. Find the velocity function. In general, is the velocity constant in magnitude or direction along the path?arrow_forwardUse Green's Theorem in the form of this equation to prove Green's first identity, where D and C satisfy the hypothesis of Green's Theorem and the appropriate partial derivatives of f and g exist and are continuous. (The quantity ∇g · n = Dng occurs in the line integral. This is the directional derivative in the direction of the normal vector n and is called the normal derivative of g.)arrow_forward(5) Let ß be the vector-valued function 3u ß: (-2,2) × (0, 2π) → R³, B(U₁₂ v) = { 3u² 4 B (0,7), 0₁B (0,7), 0₂B (0,7) u cos(v) VI+ u², sin(v), (a) Sketch the image of ß (i.e. plot all values ß(u, v), for (u, v) in the domain of ß). (b) On the sketch in part (a), indicate (i) the path obtained by holding v = π/2 and varying u, and (ii) the path obtained by holding u = O and varying v. (c) Compute the following quantities: (d) Draw the following tangent vectors on your sketch in part (a): X₁ = 0₁B (0₂7) B(0)¹ X₂ = 0₂ß (0,7) p(0.4)* ' cos(v) √1+u² +arrow_forward
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