Using Different Methods In Exercises 19-22, find
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Multivariable Calculus
- Using Properties of the Derivative In Exercise 26, use the properties of the derivative to find the following. (a) r′(t) (b) d dt [u(t) − 2r(t)] (c) d dt [(3t)r(t)] (d) d dt [r(t) ∙ u(t)] (e) d dt [r(t) × u(t)] (f) d dt [u(2t)] 26. r(t) = sin ti + cos tj + tk, u(t) = sin ti + cos tj + 1 t k * only d ,e, f *arrow_forwardGradient fields Find the gradient field F = ∇φ for the following potential function φ. φ(x, y, z) = (x2 + y2 + z2)-1/2arrow_forwardGradient fields Find the gradient field F = ∇φ for the following potential function φ. φ(x, y, z) = ln (1 + x2 + y2 + z2)arrow_forward
- Algebra find the laplace transform of x(t) = sin(t) [u(t) - u(t-2π)] h(t) = [u(t+2π)- u(t)] determine: y(t) = x(t) * h(t) sketch the grapharrow_forwardGradient fields Find the gradient field F = ∇φ for the potentialfunction φ. Sketch a few level curves of φ and a few vectors of F. φ(x, y) = x + y, for | x | ≤ 2, | y | ≤ 2arrow_forwardWork in force fields Find the work required to move an object in the following force field along a line segment between the given points. Check to see whether the force is conservative. F = ⟨x, y, z⟩ from A(1, 2, 1) to B(2, 4, 6)arrow_forward
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- Showing Linear Independence In Exercises 27-30, show that the set of solutions of a second-order linear homogeneous differential equation is linearly independent. {eax,xeax}arrow_forwardProof Use the concept of a fixed point of a linear transformation T:VV. A vector u is a fixed point when T(u)=u. (a) Prove that 0 is a fixed point of a liner transformation T:VV. (b) Prove that the set of fixed points of a linear transformation T:VV is a subspace of V. (c) Determine all fixed points of the linear transformation T:R2R2 represented by T(x,y)=(x,2y). (d) Determine all fixed points of the linear transformation T:R2R2 represented by T(x,y)=(y,x).arrow_forwardCalculus Let B={1,x,sinx,cosx} be a basis for a subspace W of the space of continuous functions and Dx be the differential operator on W. Find the matrix for Dx relative to the basis B. Find the range and kernel of Dx.arrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage LearningAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage