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**Q1:**. Sketch and find the area of the region that lies inside both curves. r^{2}=\sin (2 \theta) \text { and } r^{2}=\cos (2 \theta)See Answer**Q2:**Determine the vorticity field for the following velocity vector: V=\left(x^{2}-y^{2}\right) \hat{i}-4 x y \hat{j} \nabla\times\overrightarrow{\mathrm{v}}=_{}_{}_{}\square\quad y\hat{\mathrm{k}} the tolerance is +/-3%See Answer**Q3:**For a certain incompressible, two-dimensional flow field the velocity component in the y direction is given by the equation v=9 x y-x^{2} y Determine the velocity component in the x direction so that the volumetric dilatation rate is zero. Denote the x-independent constant as f(y).See Answer**Q4:**A two-dimensional, Incompressible flow is given by u = - y and v = x. Show that the streamline passing through the point x = 10 and y = 0 is a circle centered at the origin. x^2 + y^2 =_________. exact number, no toleranceSee Answer**Q5:**The stream function for an incompressible, two-dimensional flow field is \psi=a y-b y^{3} where a and b are constants, b+0. Is this an irrotational flow?See Answer**Q6:**For a certain incompressible flow field it is suggested that the velocity components are given by the equationsu = 14xy v = -x²y w = 0Is this a physically possible flow field? O YesO NoSee Answer**Q7:**(1 point) Let L be the line in R^3 that consists of all scalar multiples of the vector \left[\begin{array}{l} 1 \\ 2 \\ 2 \end{array}\right] Find the orthogonal projection of the vector \vec{x}=\left[\begin{array}{l} 7 \\ 3 \\ 5 \end{array}\right] \text { onto } L \text { . } \operatorname{proj}_L\vec{x}=\mleft\lbrack_{---}\mright\rbrackSee Answer**Q8:**a) Solve the differential equation (x+1) \frac{\mathrm{d} y}{\mathrm{~d} x}+x=1 \quad(x>0) b) Use separation of variables to find the solution to the differential equation \frac{\mathrm{d} y}{\mathrm{~d} x}+\sin ^{2} y=1 O when x1. Write your solutionthat satisfies the boundary condition yin explicit form for y.%3D (c) Use an integrating factor to find the general solution to the linear differentialequation x^{2} \frac{\mathrm{d} y}{\mathrm{~d} x}-x y=1 \quad(x>0)See Answer**Q9:**\text { Evaluate } \int_{C}((x-y) \mathrm{d} x+(y-x) \mathrm{d} y) \text { along the following paths } C (a) x = 1 – t², y = t between t = 0 and t = 1; b) the straight line joining (0, 0) and (1, 2); c) unit circle.See Answer**Q10:**(a) Using the first shift theorem when necessary (see below for a statement),write down the Laplace transforms of: f_{1}(t)=1-t^{3}, \quad t \geq 0 \text { ii. } f_{2}(t)=\left(1-t^{3}\right) e^{-t / 2}, \quad t \geq 0 and hence find the Laplace transform of: g(t)=\left(1-t^{3}\right)\left(1-2 e^{-t / 2}\right), \quad t \geq 0 Find constants a and b for which the inverse Laplace transform of F(s)=\frac{s+1}{2 s^{2}+\frac{1}{2}} is f(t) = a cos(t/2) + b sin(t/2), t > 0.See Answer**Q11:**-5 -2Let A be the matrix, A =(-5/3,-2/0) (a) Show that{ 2 -3 } is an eigenvector of A and find the correspondingeigenvalue. (b) Find the other eigenvalue and a corresponding eigenvector. (c) Using the results in a) and b), or otherwise, solve the vector-matrix differen-tial equation given that x(0) = {1 -2}.See Answer**Q12:**\text { Evaluate Evaluate } \int_{0}^{2} \int_{2-x}^{2}\left(2 x^{2}-3 y+1\right) \mathrm{d} y \mathrm{~d} x Sketch the region of integration. Reverse the order of integration, but do not re-evaluate.See Answer**Q13:**Problem 3- Using finite difference method, calculate the angular displacement 0(t) in the first three steps of movement with a timestep of At = 0.025 See Answer**Q14:**Sketch and set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. Identify whether the solid is disk or washer. y=\tan x, \quad y=x, \quad x=\frac{\pi}{3} ; \quad \text { about } \quad y=-1See Answer**Q15:**3. (a) Sketch and fully describe the surface Sph given, in terms of Cartesian coordinates (x, y, z), by Sph = {(x, y, z) : (x − xo)² + (y − yo)² + (z − zo)² = a²}, where a, xo, yo and zo are constants, and show that it can be parameterized by r(0,0) = (xo + a sin cos , yo + a sin sind, zo + a cos 0), giving the ranges of the parameters and p. (b) For the surface Sph, defined in part (a), show that a vector surface element is given by ds = a sin 0 [r - (To, yo, 2o)] dedo, and justify the sign convention associated with this normal vector. Hence evaluate the flux ƒ= $₁₂ Sph G.dS of the vector field G = (1,0, z²) out of the surface. (c) Using your answer to part (b), evaluate L = lima-03f/(4πa³). (d) State the (coordinate-invariant) integral form of the definition of divergence. By referring to your answer to part (c), demonstrate that it is consistent with the Cartesian definition of divergence in this case. (e) By applying the Divergence Theorem to the vector field u = ow, where is an arbitrary differentiable scalar field and w is a constant vector, derive the identity JI føds, where the closed surface S is the boundary of the volume V. VodV=See Answer**Q16:**B = 5.0 53° A = 10.0 30° ter D 37° C = 12.0 60° D = 20.0 ILL 10 30⁰ F = 20.0 Given the vectors in the preceding figure, find vector R that solves equations (a) D + R = F and (b) C-2D + 5R = 3F. Assume the +x-axis is horizontal to the right.See Answer**Q17:**Problem 1. Prove the Cauchy-Schwartz Inequality for any pair of vectors ū, v: |uv|≤|| . Hint: Recall some basic properties of the sine and cosine functions.See Answer**Q18:**Problem 2. Let e be a fixed vector in R². Describe in geometric terms the set of vectors p such that ||pc|| = 1. Then, do the same, but for a fixed vector in R³. Hint: The quantity ||p-c=1 will represent a simple shape you are familiar with.See Answer**Q19:**Problem 3. Determine the value(s) of a such that = ai + (a − 1)3 + 3ak is a unit vector.See Answer**Q20:**2 Problem 4. Earth is at the origin (0,0), the moon is at (384,0), and a spaceship is at (280,90) (where distance is in thousands of kilometers). (a) What is the displacement vector of the moon relative to Earth? Of the spaceship relative to Earth? Of the spaceship relative to the moon? (b) How far is the spaceship from the earth? From the moon? (c) The gravitational force on the spaceship from the earth is 461 newtons, and from the moon is 26 newtons. What is the net gravitational force on the spaceship?See Answer

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