10. Using Stoke's theorem, evaluate y dx+ xz dy+x² dz where C is the boundary of the triangle cut from the plane x+y+z=1 by the first octant, counter-clockwise when viewed from the above.

Q: 2. Find the distance between the line with parametric equations ï = 1 + 2t, y = −2+ 4t, z = 4 − 4t…

A:

Q: Write a paragraph explaining how the distance formula giving the distance between two points (x1,…

A:

Q: a) Use polar coordinate to evaluate I √x² + y² dA R where R is the region bounded by the circle z² +…

A:

Q: Fin He ara at tHe surface af the portion oF Hae Plane 2= 5-2X -9y

A:

Q: x + y+ 3z = 0 x-y-z%3D0 2. Find the included angle between the line and the planex -y-z+1 0.

A: we have equation of lines:    x+y+3z=0..... (i)-x+y+z=0..... (ii)Add equation (i) and…

Q: Use Green's Theorem to evaluate (V1+r) dr+2ry dy, where C is the triangle with vertices (0, 0),…

A:

Q: Use Green's theorem to evaluate O -x²ydx + x°dy, where C is the circle x? + y? = 4 with a…

A:

Q: Let F (x, y) = 8Q A. Compute 82 yi+zj z² + y² and let C be the circle r(t) = (cost)i + (sin t) j. 0…

A:

Q: Use Green's Theorem to evaluate the integral (x² - y)dax + 3xdy where C is the circle x² + y² = 1…

A:

Q: Verify Stokes' theorem for the helicoid (1, 0) = First, compute the surface integral: = (VF)…

A: Given surface is helicoid whose vector parametric equation is \)" data-mce-style="cursor:…

Q: xt=2r? yle)=r²–4¢ a) For -5sts5 , find all points (x, y) where the slope of this curve is zero.…

A:

Q: Evaluate  ∫C (3y + exsinx) dx + (8x - ln(y3+2)) dy, where C is the circle x2 + y2 = 9  with positive…

A:

Q: Let F(x, y) = = 8Q A. Compute dz -yi+zj z² + y² ар B. Compute By and let C be the circle r(t) =…

A: Given function:- Fx,y=-yi+xjx2+y2 and C be the circle rt=costi+sintj

Q: c. Evaluate f 3x ds, where C consists of arc C₁ of parabola y = x² from (0,0) to (2,4), followed by…

A:

Q: A rectangle ha verices at P (6, 6), Q (6, -6), R (-6, -6) and S (-6, 6). The origin is the center of…

A:

Q: b) Find the angle between the surfaces x² + y² + ² = 19 and 2 = 2 x² + y²-7 at the point (3, 1, 3):

A:

Q: Use hyperbolic functions to parametrize the intersection of the surfaces x² - y² = 4 and z = 5xy.…

A: Use hyperbolic functions to parametrize the intersection of the surfaces  and (Use symbolic notation…

Q: Prove that a line intersects a paraboloid at' two points which may be real or imaginary.

A: A paraboloid is a quadric surface that has exactly one axis of symmetry and no center of…

Q: Solve ut + cux=et, u(x,0) = u(x). Draw the characteristics in the x-t plane.

A: