1- The velocity field is given by V = (x²y – xy²)ï – y³ – x?)j+ xk. Find the circulation around the closed curves shown in figure 1. y |(2,4) (4,4) (2,2) (4,2) Figure 1

Q: - 11) Calculate the Jacobian, J, for the change of variables x = u cos(0) – v sin(e) and y = usin(0)…

A: The objective of this question is to calculate the Jacobian of the transformation from the variables…

Q: Let F(x, y) = (cos(x*) -v°)i + x*j. F• dr, where C is a curve consisting of the line segment joining…

A:

Q: - 11) Calculate the Jacobian, J, for the change of variables x = u cos(0) – v sin(e) and y = usin(0)…

A: The objective of this question is to calculate the Jacobian of the transformation from the variables…

Q: 3. Integrate the scalar function f(x, y, z) = √2 along the curve in R³ which goes from (0,0,0) to…

A:

Q: (a) Let f(x, y, z) be a function, let r(t) = (x(t), y(t), z(t)) be a parametric curve, and set g(t)…

A: Let the function be fx, y, z and the parametric curve be rt=xt, yt, zt. The derivative is r't=dxdt,…

Q: wy ad let (s, 1-45 24. Suppose you have a function f(x, y) and a curve r(t). You know: Vf(x, y) =…

A:

Q: - 11) Calculate the Jacobian, J, for the change of variables x = u cos(0) – v sin(e) and y = usin(0)…

A: The objective of this question is to calculate the Jacobian of the transformation from the variables…

Q: A two-dimensional flow field is given by u = 5a? – 5y, v = -10ry (i) Find the streamfunction and…

A: Given u=5x2-5y2v=-10xy

Q: Find the curl and divergence of F(r, y, z) = (e**, -2e"², 3.xe"). Please show steps/explain

A:

Q: Given that f(y) is as sketched below, find the flow vector (x, y) and sketch by hand the flow field…

A: As per the question we are given the following nonlinear dynamic system : dy/dx = f(y) And the graph…

Q: 11. Find the circulation of F(x,y) =(x-y)i+xj around the unit circle centred on the origin. y

A:

Q: -yi + xj 2. Find the flux of F across the curve C given by r(t) = a cos(t)i+a sin(t)j, VI² + y² for…

A: F=-y i+x jx2+y2,    r(t)=a cos(t) i +a sin(t) j 0≤t≤2π x(t)=-a sin(t) ,     y(t)=a cost    ,  F=-y…

Q: 1. Let F(x, y) = (y, 2x2). Use the circulation form of Green's theorem to calculate the circulation…

A:

Q: Show that the given curve c(t) is a flow line of the given velocity vector field F(x, y, z). c(t) =…

A: We have to show the following conditions according to the given information

Q: Suppose F(x, y) = es sin(x)] and C is the counter-clockwise oriented rectangle with vertices (0, 0),…

A:

Q: 2. Calculate the circulation of the flow v=-8xy i+ 4 yj around the triangle with vertices (0, 0),…

A: Given that: v=-8xyi+4y2j The triangle is with the vertices (0,0), (3,0), (3,2) To calculate the…

Q: 3. Given function z = f(x,y) = x²y² – ry + 2x – 2y (a) Find the differential dz. (b) Find the…

A: The differential can be obtained using the formula dz=fxdx+fydy. Here, fx, fy are derivative with…

Q: Let w = x°e4y cos(9z). Find the value of dwldt at the point (3, ln 5, 0) on the curve x = 3 cos(t),…

A: Given: w=x6e4ycos(9z)x=3 cos(t)y=ln(t+5)z=6tDifferentiating we…

Q: yi + aj 2. Find the flux of F = across the curve C given by r(t) = a cos(t)i+a sin(t)j, ²+y² for 0…

A: ∵ Given thatF=-yi^+xj^x2+y2;           0≤t≤2π r(t)=acosti^+asintj^ As we know that Flux=∬ F·nds…

Q: 3. Use integration to determine the formula for the "equatior: given that : F(t) = ( 2t , 12e" , sin…

A:

Q: 2. Let v(x, y) = (2). (a) Show that v(x, y) is an ideal flow. (b) Find the complex potential Þ(z)…

A: d)

Q: - 11) Calculate the Jacobian, J, for the change of variables x = u cos(0) – v sin(e) and y = usin(0)…

A: The objective of this question is to calculate the Jacobian of the transformation from the variables…

Q: Find the flux of material as a function of r and t and show that at z = 1 the material is tranferred…

A: As per the question, we are given the material density as a function of x and t, and we are asked to…

Q: Suppose F(x, y) = (4x - 3y)i + 2xj and C is the counter-clockwise oriented sector of a circle…

A:

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.