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- Let T be a linear transformation from R2 into R2 such that T(x,y)=(xcosysin,xsin+ycos). Find a T(4,4) for =45, b T(4,4) for =30, and c T(5,0) for =120.arrow_forwardLet x=x(t) be a twice-differentiable function and consider the second order differential equation x+ax+bx=0(11) Show that the change of variables y = x' and z = x allows Equation (11) to be written as a system of two linear differential equations in y and z. Show that the characteristic equation of the system in part (a) is 2+a+b=0.arrow_forwardsuppose Vn is the vector space of polynomials of degreearrow_forward6 Find the curl of the vector field F = < yx³, xz5, zy curl F= + S karrow_forwardThe theory of complex variable comes in naturally in the study of fluid phenomena.Letus define, w = φ + iψ with φ velocity potential and ψ stream function of a two dimensional fluid flow. w is called the complex potential function.Then w turns out to be analytic because the Cauchy-Riemann conditions, ∂φ/∂x =∂ψ/∂y ; ∂φ/∂y = −∂ψ/∂x are exactly the natural flow conditions that have to be satisfied.Suppose, velocity potential for a two dimensional fluid flow is given by the functionφ(x, y) = y/x2+ y2− 2xy.Find out the stream function. Also find the complex potential function for the fluid flow.arrow_forward(a) Prove that the vector field F(x, y, z) = (x² + yz)i – 2y(x + z)j + (xy + z²)k is incompressible, and find its vector potential function. ||r. If f is a differentiable function of one (b) Let r = ri + yj + zk, and let r variable, show that ▼ · (ƒ(r)r) = rf'(r) +3ƒ(r) (c) If o and u are smooth scalar fields, show that ▼ × (6V) = ▼¢ × Vv Vox (d) Let f(1, y) = 2r² + xy - y². Prove that the directional derivative of f(x, y) at point x = (3,-2) in the direction v=i-jis.arrow_forwardarrow_back_iosarrow_forward_ios
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