1)calculate a*yz dædydz where D is the rectangular parallelepiped 1< « < 2,1 < y < 2,1 < z < 5

Q: Evaluate the line integral [6. 6ayⓇ ds, where C is the right half of the circle x² + y² = 4.

A: To evaluate :   ∫C6xy6ds, where C is the right half of the circle x2+y2=4 Prerequisite : We can…

Q: |Use Green's Theorem to evaluate [(3x² +y)dx+4xydy where C is the triangular C region with vertices…

A: The solution is given as

Q: Let Li be the straight line segment from (0.0,1) to (1,a, 8) and let F: (z+ax, ay, x> Calculate the…

A:

Q: Prove that the parabola y = x^2 at the points where the line has an angle of inclination "o" the…

A: Curvature of  a graph y= f(x)  k=d2ydx21+dydx232

Q: Exercise 8. Show that the sum of the angles of a geodesic triangle on a surface of positive…

A: We will use the Gauss–Bonnet theorem to prove the given statement.

Q: Show by Vector mothads that th ditonce hom fhe fornt P.(Xx, Y,, Z) to the plane Ax -+By+cz-D26 t of…

A:

Q: Calculate the line integral ∫Cy2ds, where C is a part of the circle x=acost, y=asint, 0≤t≤π/2.

A:

Q: 4. Does there exist a simple closed unit-speed plane curve with curvature less than or equal to 2…

A: In a simple closed unit-speed plane curve, curvature =1r According to the question, Area of the…

Q: State the equstion of the yz plane

A: We have to find equation of yz plane

Q: 3) Prove thet the xy-plane with constant curva tue k= urve In is o circle of radious

A: We have to prove that a curve in the xy plane with constant curvature k=1r is a circle of radius r

Q: 2. Given: WX = YZ; ZWXZ = ZYZX Prove: WXYZ is a parallelogram

A: Parallelogram

Q: 4. Show that the set of points (z, w) e C with w2 = sin z is naturally a Riemann surface. %3D

A: Let, S=z,w∈ℂ2|w2=sinz be the set in equation. The multivalued analytic function fz=sinz can be…

Q: Use Green's theorem to evaluate $(xy + 3x²)dx + (3y – 2x)dy, along C, where Cis the triangle with…

A: Given Integral ∮(xy+3x2)dx +(3y-2x)dy along c where c is the triangle with vertices (0, 0), (0, 3)…

Q: Consider points P and Q in the x-y plane, with P = (1,0) and Q = (0,1). The line integral %3D Q…

A: The line joining P1,0 and Q0,1 is     x-11-0=y-00-1=t(say)⇒x=t+1,y=-t⇒dx=dt,dy=-dt

Q: Evaluate the line integral / x°z ds, where C is the line segment from (0, 2,4) to

A: Line integral: - The function to be integrated is determined along a curve in the coordinate system…

Q: Consider 2-dimensional hyperbolic space with metric ds ds'+dy and y > 0. The length of a line…

A:

Q: Show that, in the triangular region with vertices (0,0), (1,0) and (1,1) the following double…

A: Double integral

Q: WX || ZY and Zs Z and Y are supplementary WXYZ is a parallelogram Given: Prove: W Y

A: It is given that,  WX ¯|| ZY¯  and ∠Z and ∠Y are supplementary To prove :- WXYZ is a parallelogram

Q: Let C be the curve which is the union of two line segments, the first going from (0, 0) to (4, 3)…

A:

Q: 7. Which surface is parameterized by ř(u, v) = (u, uv, v) for –1 < u <1 and –1< v< 1?

A: r→(u, v) = u, uv, v for -1≤u≤1 and -1≤v≤1The parametric curve is  Option C is correct.

Q: 2-Show that the regular curve a = a(s) is a straight line if and only if its curvature k is…

A: Here we have to show that a regular curve is a straight line if and only if, its curvature is…

Q: Find vohume fome above ZVy-xLy, below Z0(xy plane) inside xy/ 3.

A:

Q: Compute where Cis the counter clockwise oriented boundary of upper-half unit disc D. 0.66666666667…

A: We need to find ,  ∮ y2dx + 3xydy , over  C ,  where C is the counter clockwise oriented boundary of…

Q: Let C be the curve which is the union of two line segments, the first going from (0, 0) to (3, 2)…

A: C be the curve point (0,0) to (3,2) and (3,2) to (6,0) so, integral we need to compute I =∫c3dy-2dxI…

Q: consider surtace S with paramtrization. Gcu,v) = cu,V,uv) , tuts of s also compute the surface areu

A: Given that  G(u,v) = &lt;u,v,uv&gt; , u2+v2≤1 Surface area is given by, S=∫∫D|Gu×Gv|du dv To find Gu…

Q: 4) Find the families of lines of curvature of the surface z = r + y? .

A: Given: z=x2+y2

Q: The tangent plane to the elliptic paraboloid z = 2x2 + y² at the point (1, 1, 3) is %3D 1) 8x + 6y +…

A:

Q: Show thata perojective plane of gives 0nder indeed BİBD (+n+l, n+),1)

A: We need to show that the projective plane of order n indeed gives a BIBD (n2+n+1,n+1,1)

Q: Find the Largest open rectangle in the plane in which the hypotheses of Existence and uniqueness…

A: Answer

Q: Calculate fc((3x + 5y) i + (3x + 4y) j) - dr where C is the circular path with center (a, b) and…

A: According to the greens theorem:

Q: Let C be the line segment from point (0, 1, 1) to point (2, 2, 3). Evaluate line integral yds. C

A: To evaluate the line integral

Q: Let P be the plane 2x + y – z = 2 and let Q(1,0,3) be a point. We want to find the minimum distance…

A:

Q: Suppose a point X lies in the exterior of a disk, and suppose that two lines are drawn from X…

A:

Q: Just like in a projective plane, two distinct lines always intersect on the spherical plane.

A: Straight lines in spherical plane are the great circles. So, any two different lines meets at two…

Q: 28. Show that the ellipsoid 2x² + 3y² + z² = 9 and the sphere %3D x? + y? + z? - 6x – 8y – 8z + 24 =…

A:

Q: 4 6 Let y = 7 2 40 Compute the distance d from y to the plane in IR spanned by ui and uz d =

A:

Q: Find the distance between the planes 4x + 5y + z = 25 and 4x + 5Y + z = 53. %3D

A: Explanation of the solution is given below....

Q: Let C be the positively oriented square with vertices (0,0), (3,0), (3,3), (0,3). Use Green's…

A:

Q: Consider 2-dimensional hyperbolic space with metric ds = dr'+dy and y > 0. The length of a line…

A:

Q: . By sketching the traces on the ry-, xz- and yz-plane

A:

Q: Calculate ff A.n ds If A = yi + 2xj – zk and S is a 2x + y = 6 plane surface in the first octane cut…

A:

Q: Sketch the reglon enclosed by the given curves. Dedde whether to iIntegrate with respect to or y.…

A:

Q: (1) Compute where Cis the couater dockwise oriented boundary of upper half unit disc D.

A: It is given that, ∮Cy2dx+3xydy here C is counter clockwise oriented upper half disc D.   Here,…

Q: 4) Find the families of lines of curvature of the surface :=r+y

A: The answer is given in handwritten format --

Q: 4. Suppose that VxF= (0,1,-1), S is the plane x + y + z = 1 in the first octant and Cis the boundary…

A: Given: ∇×F=01-1 S is the plane x+y+z=1 in the first octane  and C is the boundary of S. Now, to find…

Q: 3. Show that every sphere through the circle x2 +y-2ax+r²=0, z=0 cuts orthogonally every sphere…

A: This is a co-ordinate geometry problem

Q: Calculate (x+ y)dA, where Ris a triangular region with vertices (0,0), (3,0) and (3,3).

A:

Q: Calculate both sides of Green's Theorem when F = [-x²y, xy²] and y positively oriented circle x? +…

A:

Q: Find a parametrization for the line segment joining points (0, 2) and (4, 0) using the angle 0 in…

A: Given: The line segment joining (0, 2) and (4, 0). Do the construction as in below figure

Q: Let S be the portion of the cylinder y = 1 − x2 with x≥0; y≥0, bounded by the planes z = 2, and z =…

A: