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Using the Fundamental Theorem of Line
F (x, y) =
C: line segment from
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Multivariable Calculus
- 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_forwardEvaluate the line integral (ci+ 3xyj – (x + z)k) · dr where C is the parametric curve r(t) = (1 – t)i + (4 + t)j + (2 – t)k, 0arrow_forward1. Consider the function F(x, y, z) = (√/1 – x² − y², ln(e² — z²)). This function is a mapping from R" to Rm. Determine the values of m and n. (b) Is this function scalar-valued or vector-valued? Briefly explain. (c) Determine the domain and range of F and sketch the corresponding regions. (d) Is it possible to visualize this function as a graph? If so, sketch the graph of F.arrow_forwarda) Find Maclaurin expression of f (x) = Cosh (x) b)Find the gradient of the function f(x, y, z) = x² + y³ – 2z + z In x at point P (2, 2,1) %3D -arrow_forwardHow to do this?arrow_forward(a) Verify that the Fundamental Theorem for Line Integrals can be used to evaluate the integral Sc ey dx + e dy, where C is the parabola r(t) =, for –1arrow_forward1. Consider the vector-valued function F(t) = sin(2t) t (2, 1, -2), a) Find the domain of F. b) Show that F is continuous at t = 0. ; et, t 1-√1+t, t #0 t = 0. = (1/2/3/17, -2). 2. Consider the vector-valued function R(t) with R(1) = (1,2,−1) and R (t) = (1 a) Find '(1) if (t) = (-2, -3t, t²) × R'(t). b) Find the equation of the normal plane to the graph of Ả at t = 1. c) Find the arc length of the graph of R from the point at t = 1 to the point at t = 3. d) Find the curvature of the graph of R at t = 1.arrow_forwardDetermine a so that u(x,y) = e-πx cos ay is harmonic and finda) the harmonic conjugate function v(x,y) of u(x,y). b) the corresponding analytic function f(z).(f = u+iv)arrow_forwardEvaluating line integral, can u add explanation to question solutionarrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_iosRecommended textbooks for you
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