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

Q: 5. Consider the surface: x(u, v) y(u, v) z(u, v) where u € [0, 1] and v € [0,2]. Calculate the area…

A: Introduction: The area is the area occupied by a two-dimensional flat surface. It has a square unit…

Q: 6. For any closed surface S, prove that || CurlF.N dS=0.

A:

Q: Identify and sketch the graph of the following SURFACES. x² + y² = sin²z, z = [0, π]

A: Any point on the 3D plane is represented using the ordered pair: x, y, z. A graphing utility can be…

Q: Identify and sketch the graph of the following SURFACES. x² + y² = sin² z, Z = [0, π]

A: Given.   x^2+y^2=sin^2z

Q: Determine the points on the surface y2+z2−x=8 that are closer to the origin.

A: Consider the given surface. y2+z2−x=8z2=8−y2+xf(x,y)=8−y2+x

Q: Find the extreme values of ƒ(x, y, z) = x - y + z on the unit sphere x2 + y2 + z2 = 1.

A:

Q: Suppose the temperature at the point (x, y, z) on the sphere x2 + y2 + ?2 = 16 is T= 800xy2z3 .…

A: Solution is given below:

Q: 28. Show that the circulation of F(x, y, z) = (x², y², z(x² + y²)) around any curve C on the surface…

A: Given figure Stoke's Theorem Here take single integral over closed path C and double integral over…

Q: the surface z = &x + 4y th=

A:

Q: 5. Find the area and centroidal distances of the curve that lies between he equations y%3D4-x^2 and…

A: The region bounded by curves y=4-x2andy=x+2 To find the area and centroidal distances.

Q: Find the area of the resulting surface if an arc of a parabola y = x² from (1,1) to (2,4) is rotated…

A: Given

Q: Find the points on the surface x2 - zy = 4 closest to the origin.

A:

Q: 17. Find the exact area of the surface z = 1 + 2x + 3y + 4y2, 1x 4, 0 y < 1

A:

Q: 14) Find the points on the surface xy- z2 = 1 that are closed to the origin. %3D

A: To find the point on the below surface that is closed to the origin. xy−z2=1

Q: 4. Graph the surface 1 = y² + z² – by first sketching at least three cross sections in each of the…

A: Graph the surface by using a 3D graphing calculator like desmos. Three cross sections in xy plane…

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: 2-By revolving the semicircle: X= r cos t, Y=r sin t, (0<t<n) About the x-axis, Show that the…

A:

Q: Find the equation of the sphere of radius 8 centered at (1, 3, 6). O x² + y² +z² = 46 O x² + y² + z²…

A:

Q: If (a,B,y) is a point at which the surface x-+y²-z²-2x+12=0_has a horizontal tangent plane, then…

A:

Q: 3. Find the area of the surface generated by revolving the curve y = 2/x ,1Sx<2, about the x-axis(as…

A:

Q: If the infinite curve  y = e−4x, x ≥ 0,  is rotated about the x-axis, find the area of the…

A:

Q: 1. Find the closest point of the surface ry + xz + yz = 1 to the origin. %3D

A:

Q: 1. Find the smallest distance from the point (0, c, 0) to the surface y = x² + z².

A:

Q: 12. Find the area of the surface generated when the curve y = ,on [1,2] is rotated about the x-axis.

A:

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: 4. If the curve y = 3 – x², 0<x<4 is rotated about the y-axis, find the area of the resulting…

A:

Q: Consider the parametric surface S given by the portion of the sphere x +y² +z² = 4 where z 2 1.

A: In this question, we have to parameterize the surface in cartesian coordinates and then find the…

Q: ded above by surface z = 10 – x2, bou e planes y = 0 and y = 3.

A:

Q: 10. Find the area of the paraboloid x² + y + z² = 2 by the plane y = 0. (Hint: Implicit Surface)

A:

Q: 9. Show that the sum of the T-, y-, and z- intercepts of any tangent plane to the surface VI + Vỹ +…

A: Given, Gradient of the surface at any point (x0,y0,z0) is given by,

Q: If the infinite curve  y = e−8x, x ≥ 0,  is rotated about the x-axis, find the area of the…

A: Given: y=e-8x , x≥0

Q: If a parametric surface given by r1(u, v) = f(u, v)i + g(u, v)j+ h(u,v)k and –1< u<1, –2 < v< 2, has…

A: We will find out the required value.

Q: The tangent plane to the graph of the surface z=In(x2+y2) at the point (1,0,0) contains point…

A: Given function is, z=lnx2+y2

Q: Graph the surface 1 = y? +z? - by first sketching at least three cross sections in each of the xy…

A:

Q: Find the points on the surface xy + yz + zx - x- z =0 where the ant pla

A:

Q: If (a,B,y) is a point at which the surface x²+y²-z²–2x+104=0 has a horizontal tangent plane, then…

A:

Q: Find the points on surface z^2=xy+4 closest to origin.

A:

Q: Evaluate [[[ =*dxcydz over the sphere x +y² +=² =1.

A: to find:- to evaluate the given integral.

Q: Suppose that the Celsius temperature at the point (x, y, z) on the sphere x2 + y2 + z2 = 1 is T =…

A: We have given; The temperature T at any point (x, y, z) in space isT = 400xyz2and the unit spherex2…

Q: Let M be the surface given by z = y^2 − x^2. Compute its Gaussian and mean curvatures at all points

A:

Q: Suppose that the Celsius temperature at the point (x,y,z) on the sphere x +y +z =1 is T= 1600xyz“.…

A: This is a problem related to maxima and minima. Based on the general formula we will answer the…

Q: 12. Find the area of the surface generated when the curve y =+,on [1,2] is rotated about the x-axis.

A: solution is given below...

Q: Determine the area bounded by the conic sections y = x² – 6² and y = 6² – 2². O 576 O 183 O 281 O…

A:

Q: Find the points on the surface xy – z2 = 1 that are closed to the origin.

A: Given surface is xy-z2=1. let xy-z2-1=0⇒z2-xy+1=0 Now distance from origin is…

Q: 2) Find the area of the part of the surface z = x2 + 2y that lies above the triangle with vertices…

A:

Q: 2. Find the area of the surface obtained by rotating the curve: y = Vx +1,1<x<5 about the Х-ахis

A:

Q: 7. Find the area of the portion of the plane z = 2y that lies directly above the circle x +(y-1)² =1…

A: we have to find area of the portion of plane .and coverting the problem in polar coordinate makes it…