Cauchy-Riemann Equations In Exercises 105 and 106, show that the functions u and v satisfy the Cauchy-Riemann equations
Want to see the full answer?
Check out a sample textbook solutionChapter 13 Solutions
Calculus: Early Transcendental Functions
- Interpreting directional derivatives Consider the functionƒ(x, y) = 3x2 - 2y2.a. Compute ∇ƒ(x, y) and ∇ƒ(2, 3).b. Let u = ⟨cos θ, sin θ⟩ be a unit vector. At (2, 3), for what values of θ (measured relative to the positive x-axis), with 0 ≤ θ < 2π, does the directional derivative have its maximum and minimum values? What are those values?arrow_forwardPractie go7 Let ū = [1, 1, 1), ở = [2, 2, 4]. (a) Find i· T. (b) Find i x ū. (a) Show that ū xở is perpendicular to d.arrow_forwardTensorial Calculus Convert the differential equation to polar (using the fact that ∇f is a covariant vector) and solve for f(x, y). Please dont skip stepsarrow_forward
- Vector-valued functions (VVF) VVF Point P t value (for point 1 r(t) = ½ t²i + √4 − t j + √t + 1k (0, 2, 1) Graph the tangent line together with the curve and the points. Copy your code and the resulting picture. Use MATLAB (Octave) code for graphing vector valued functions. A) 2arrow_forwardLet F(t) = (31³-3, 4et, -sin(4t)) Find the unit tangent vector T(t) at the point t = 0 T(0) = < 0 Question Help: Add Work " Videoarrow_forwardGravitational potential The potential function for the gravitational force field due to a mass M at the origin acting on a mass m is φ = GMm/ | r | , where r = ⟨x, y, z⟩ is the position vector of the mass m, and G is the gravitational constant.a. Compute the gravitational force field F = -∇φ .b. Show that the field is irrotational; that is, show that ∇ x F = 0.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_forwardRepresent the plane curve by a vector-valued function. x² = y2 = r(t) = 2 cos t + sin t j Need Help? Read It Watch Itarrow_forwardDetermine the type of points on the X (u, v) = (u, v, u?) surface. Differential geometryarrow_forward
- Differentiation of Vector-Valued Functions In Exercises 7 and 8, find r (t), r(t,), and r (t,) for the given value of t. Then sketch the space curve represented by the vector-valued function, and sketch the vectors r(t) and r'(t). 7. r(t) = 2 cos ti + 2 sin tj + tk, toarrow_forwardDerivatives of vector-valued functions Differentiate the following function. r(t) = ⟨4, 3 cos 2t, 2 sin 3t⟩arrow_forwardAngular speed Consider the rotational velocity field v = ⟨0, 10z, -10y⟩ . If a paddle wheel is placed in the plane x + y + z = 1 with its axis normal to this plane, how fast does the paddle wheel spin (in revolutions per unit time)?arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageAlgebra and Trigonometry (MindTap Course List)AlgebraISBN:9781305071742Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage LearningElementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning