(x, y)Rz + z = x V z = y (x, y)Sz → (x – y²)² + (z² – æ²)² = 0

Q: Consider the function f(x, y) = In(x² + y² + 1). 2 (a) Find Vf(1, 2). (b) Find Daf(1, 2) in the…

A: f(x,y)=ln(x2+y2+1) (a)Gradient vector…

Q: T(x,y, z) = x² + 2y² + z

A:

Q: F(x, y,z)= (4xyz) î + (-2xz+2yz) ĵ -3y² z k , find VxF

A:

Q: Solve for u(x, t): 1 3ux + 4u=0, u(x, 0) = 1 + x²

A: Given that the differential equation is:                             To find the solution of the…

Q: Evaluate (xy + yz + xz)dV B = {(x, y, z) | 0 < x < 2,0 < y < 2,0 < z < 4}

A:

Q: Given f(x, y) = x° + 2xy³ – 2y", find fz(x, y) = fy(x, y) =

A: Given  f(x, y) = x6 + 2xy3 - 2y5.

Q: Compute the discriminant D(x, y) of the function. ƒ(x, y) = x³ + y4 − 6x − 2y² + 2

A:

Q: Find fa, fy, fz. f(x, y, z) x²y 2z+1

A:

Q: Find [2² Pex. 2 √²+ 3 1³ F(x f(x, y) = 5x + 3x²y² 3 √³ F(x, f(x, y) dx and f(x, y) dx = f(x, y) dy =…

A:

Q: dy If y (t+ 2t + 6) (7t + 6), r find dt dy dt

A: Given  y=t2+2t+67t2+6y=7t4+6t2+14t3+12t+42t2+36y=7t4+14t3+48t2+12t+36

Q: Calculate r' (t) and T(t), where 1668 T(t) = r(t)= (4t-4, 1- 4t, 6t – 1).

A:

Q: Given f(x, y) = 2x² + 6xy² + y³, find frz(x, y) = fry(x, y) =

A:

Q: f) Given w=(x+y)/(z-3y) , Find dw

A: Total derivative

Q: f (x, y) = 3y + xy – a² – y? Find Vf.

A:

Q: Compute the discriminant D(x, y) of ƒ(x, y) = x2y2.

A: fx, y=x2y2 Partially differentiating with respect to x, we get ∂f∂x=2xy2 Partially differentiating…

Q: Find r'(t), r"(t), r'(t) · r"(t), and r'(t) x r"(t). r(t) = 2t²1 - 4tj + (a) r(t) (b) r"(t) (c)…

A:

Q: Let f(x, y) = 2x² + 4y² + 2xy+ 10x + 3y + 9 Find D = frr · fyy - i

A:

Q: f(x, y) =  5 xy +  xy 8; find fx, fy, and fyy

A: Partial derivatives

Q: Calculate r'(t) and T(t), where r(t) = (2t – 5, 4 – 4t, 1 + 4t). %3D r'(t) = { %3D T(t) = {

A:

Q: Find r'(t), r"(t), r'(t) · r"(t), and r'(t) x r"(t). r(t) = 5t²i - 5tj + 3t³k 2 (a) r(t) (b) r"(t)…

A: Note: As per the guidelines, I have solved three sub-parts. If you want the solution to the last…

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.