Let B be a solid delimited by a smooth surface S.
a) Show that the volume of B can be calculated using the formula
Let B be a solid delimited by a smooth surface S. a) Show that the volume of B can be calculat...
Let S be a smooth parametric surface and let P be starts at P intersects S at most once. The solid angle (S) subtended by S at P is the set of lines starting at P and passing through S. In this problem, we define the measure of the solid angle (in steradians) as a point such that each line that r n dS, where r is the position vector from P to any point on S, l, and the...
Let R be delimited by and and S being surface R, outwardly. Now give us the vector field F(x,y,z)=ij + calculate flux integral We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image(z + sin ( 2)) +(y + cos(r3 +(22 + sin(zy))k
surface patch for S. regular surface and f: S Ra smooth EXERCISE 3.44. Let S be a function. Assume that the point p e S is a critical point of f, which means that dfp(v) 0 for all v e TpS. Define the Hessian of f atp in the direction v as Hess(f)p(v) (foy)"(0), where y is a regular curve in S with y(0) = p and y'(0) = v. Prove that the Hessian is well defined in the sense...
Calculus . Let h(x, y) be a smooth parametrization on a region H for a surface S in R3. Suppose there is a continuous transform F :R + H, (u, v) + x(u, v), y(u, v)) such that F is one-to-one on the interior of the region R and r;=ho F is a smooth parametrization on R for S. Show that 9 S/ \ru xroldA= S/ \he x hy|dA= A(s where A(S) is the area of S. (15 pts] 9
14. (1 point) Let S be the boundary surface of the solid B Ir,v.ry:10S:S F. dš (a) directly as a surface integral AND (b) as a triple integral by uring the Divergence Theorem. Spring 2005) 14. (1 point) Let S be the boundary surface of the solid B Ir,v.ry:10S:S F. dš (a) directly as a surface integral AND (b) as a triple integral by uring the Divergence Theorem. Spring 2005)
please be as detailed as possible Question 5, Let ơ (u, v) : R2- R3 be a smooth function (not necessarily a surface patch). Let E Ou .Ou, F-Ou . συ and G Oy .Oy. Show that the following equalities hold: (Here D denotes total derivative.) Question 5, Let ơ (u, v) : R2- R3 be a smooth function (not necessarily a surface patch). Let E Ou .Ou, F-Ou . συ and G Oy .Oy. Show that the following equalities...
Define the tangent space of a Point P of a smooth Surface S and show that it is a two-dimensional veector space.
Determine the total surface area in mm2 and the volume in mm3 of the solid of revolution. (Let 50 mm be the difference between the inner and outer radii at the top of the solid.) 50 mm 45 mm 75 mm 25 mm A = mm 2 mm3
5. (6) Consider the solid bound in the first octant by the surface 9x2 +4y 36 and the plane 9x+ 4y + 6z 36. a. Sketch the solid. b. Set-up the integral to find the volume of the solid by using a double integral. DO NOT INTEGRATE 5. (6) Consider the solid bound in the first octant by the surface 9x2 +4y 36 and the plane 9x+ 4y + 6z 36. a. Sketch the solid. b. Set-up the integral to...
Let S be the surface of the solid bounded by the cylinder x ^2 + y ^2 = 9 and the double-cone z^ 2 = x ^2 + y^ 2 . Evaluate double integral <x ^3 , y^3 , cos(xy)>· dS