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5. A surface is parameterized by u and v as xacosucosv y bsinucosv z csinv Here...
Notation and convention: r x +y The distance from the origin to the point r [x,y,z]...
Notation and convention: r x +y The distance from the origin to the point r [x,y,z] + ê: The unit vector along the direction of r-[x, y,z] (a.e,6)-i.j.):m :The orthonormal bases of a Cartesian coordinate system. for dummy indices Einstein convention: Omitting the summation notation (repeated indices). Examples:ab,-a b, ab a b Notice: No dummy index is allowed to be repeated more than twice. You should change the "names" of the dummy indices before taking the product of two summations...
first picture is the notation on the test paper Second picture is the question i would...
first picture is the notation on the test paper
Second picture is the question i would like to solve
thanks~
Notation and convention: r x +y The distance from the origin to the point r [x,y,z] + ê: The unit vector along the direction of r-[x, y,z] (a.e,6)-i.j.):m :The orthonormal bases of a Cartesian coordinate system. for dummy indices Einstein convention: Omitting the summation notation (repeated indices). Examples:ab,-a b, ab a b Notice: No dummy index is allowed to be...
Consider the vector field F(x, y, z) -(z,2x, 3y) and the surface z- /9 - x2 -y2 (an upper hemisph...
Consider the vector field F(x, y, z) -(z,2x, 3y) and the surface z- /9 - x2 -y2 (an upper hemisphere of radius 3). (a) Compute the flux of the curl of F across the surface (with upward pointing unit normal vector N). That is, compute actually do the surface integral here. V x F dS. Note: I want you to b) Use Stokes' theorem to compute the integral from part (a) as a circulation integral (c) Use Green's theorem (ie...
10. Consider the surface S parameterized by w r= (cos y, sin v, u + sin...
10. Consider the surface S parameterized by w r= (cos y, sin v, u + sin v), -3 <u <3, 050 < 27 *** (a) Write a linear equation for the tangent plane to the surface at (0,1,1) (b) Compute the surface area of S.
Example A.3 Surface normal vector. Let S be a surface that is represented by f(x, y, z) -c, where...
Example A.3 Surface normal vector. Let S be a surface that is represented by f(x, y, z) -c, where f is defined and differentiable in a space. Then, let C be a curve on S through a point P-Go, yo,Zo) on S, where C is represented by rt)[x(t), y(t), z(t)] with r(to) -[xo. Vo, zol. Since C lies on S, r(t) must satisfy f(x, y. z)-c, or f(x(t), y(t), z(t))-c. Show that vf is orthogonal to any tangent vector r'(t)...
(14 points) Let F be the radial vector field Ft(z, y, z) =zi+w+sk And S be the surface of the cone shown at right parameterized by G(r,)-(rcos(0),r sin(0),6-3r) Write the integral F dS using an outwa...
(14 points) Let F be the radial vector field Ft(z, y, z) =zi+w+sk And S be the surface of the cone shown at right parameterized by G(r,)-(rcos(0),r sin(0),6-3r) Write the integral F dS using an outward pointing normal in dS terms r and θ. This cone has an open bottom. . The integrand must be fully simplified » Do not evaluate the integral
(14 points) Let F be the radial vector field Ft(z, y, z) =zi+w+sk And S be the...
Question 1 1 pts Let F= (2,0, y) and let S be the oriented surface parameterized...
Question 1 1 pts Let F= (2,0, y) and let S be the oriented surface parameterized by G(u, v) = (u? – v, u, v2) for 0 <u < 12, -1 <u< 4. Calculate | [F. ds. (enter an integer) Question 2 1 pts Calculate (F.ds for the oriented surface F=(y,z,«), plane 6x – 7y+z=1,0 < x <1,0 Sysi, with an upward pointing normal. (enter an integer) Question 3 1 pts Calc F. ds for the oriented surface F =...
Assume that is the parametric surface r= x(u, v) i + y(u, v) j + z(u,...
Assume that is the parametric surface r= x(u, v) i + y(u, v) j + z(u, v) k where (u, v) varies over a region R. Express the surface integral 116.3.2) as as a double integral with variables of integration u and v. a (x, y) a(u, v) du dy ru Хry dy du l|ru Xr, || f (x (u, v),y(u, v),z (u, v)) 1(xu, Wsx,y,z) Mos u.v.gou,» @ +()*+1 li ser(u, v),y(u, v),z (u, v) Date f (u, v,...
5. Suppose σ is a parametric surface with vector equation r(14. u) x (u, u)i + y(u, u)j + z(u, v)...
5. Suppose σ is a parametric surface with vector equation r(14. u) x (u, u)i + y(u, u)j + z(u, v)k If σ has no self-intersections and σ 1s smooth on a region R in the uu-plane, then the surface area of ơ is given by
5. Suppose σ is a parametric surface with vector equation r(14. u) x (u, u)i + y(u, u)j + z(u, v)k If σ has no self-intersections and σ 1s smooth on a region R...
(1) Let P denote the solid bounded by the surface of the hemisphere z -Vl-r-y? and the cone2y2 and let n denote an outw...
(1) Let P denote the solid bounded by the surface of the hemisphere z -Vl-r-y? and the cone2y2 and let n denote an outwardly directed unit normal vector. Define the vector field F(x, y, z) = yi + zVJ + 21k. (a) Evaluate the surface integral F n dS directly without using Gauss' Divergence Theorem. (b) Evaluate the triple integral Ш div(F) dV directly without using Gauss' Diver- gence Theorem Note: You should obtain the same answer in (a) and...
Notation and convention: r x +y The distance from the origin to the point r [x,y,z] + ê: The unit vector along the direction of r-[x, y,z] (a.e,6)-i.j.):m :The orthonormal bases of a Cartesian coordinate system. for dummy indices Einstein convention: Omitting the summation notation (repeated indices). Examples:ab,-a b, ab a b Notice: No dummy index is allowed to be repeated more than twice. You should change the "names" of the dummy indices before taking the product of two summations...
first picture is the notation on the test paper
Second picture is the question i would like to solve
thanks~
Notation and convention: r x +y The distance from the origin to the point r [x,y,z] + ê: The unit vector along the direction of r-[x, y,z] (a.e,6)-i.j.):m :The orthonormal bases of a Cartesian coordinate system. for dummy indices Einstein convention: Omitting the summation notation (repeated indices). Examples:ab,-a b, ab a b Notice: No dummy index is allowed to be...
Consider the vector field F(x, y, z) -(z,2x, 3y) and the surface z- /9 - x2 -y2 (an upper hemisphere of radius 3). (a) Compute the flux of the curl of F across the surface (with upward pointing unit normal vector N). That is, compute actually do the surface integral here. V x F dS. Note: I want you to b) Use Stokes' theorem to compute the integral from part (a) as a circulation integral (c) Use Green's theorem (ie...
10. Consider the surface S parameterized by w r= (cos y, sin v, u + sin v), -3 <u <3, 050 < 27 *** (a) Write a linear equation for the tangent plane to the surface at (0,1,1) (b) Compute the surface area of S.
Example A.3 Surface normal vector. Let S be a surface that is represented by f(x, y, z) -c, where f is defined and differentiable in a space. Then, let C be a curve on S through a point P-Go, yo,Zo) on S, where C is represented by rt)[x(t), y(t), z(t)] with r(to) -[xo. Vo, zol. Since C lies on S, r(t) must satisfy f(x, y. z)-c, or f(x(t), y(t), z(t))-c. Show that vf is orthogonal to any tangent vector r'(t)...
(14 points) Let F be the radial vector field Ft(z, y, z) =zi+w+sk And S be the surface of the cone shown at right parameterized by G(r,)-(rcos(0),r sin(0),6-3r) Write the integral F dS using an outward pointing normal in dS terms r and θ. This cone has an open bottom. . The integrand must be fully simplified » Do not evaluate the integral
(14 points) Let F be the radial vector field Ft(z, y, z) =zi+w+sk And S be the...
Question 1 1 pts Let F= (2,0, y) and let S be the oriented surface parameterized by G(u, v) = (u? – v, u, v2) for 0 <u < 12, -1 <u< 4. Calculate | [F. ds. (enter an integer) Question 2 1 pts Calculate (F.ds for the oriented surface F=(y,z,«), plane 6x – 7y+z=1,0 < x <1,0 Sysi, with an upward pointing normal. (enter an integer) Question 3 1 pts Calc F. ds for the oriented surface F =...
Assume that is the parametric surface r= x(u, v) i + y(u, v) j + z(u, v) k where (u, v) varies over a region R. Express the surface integral 116.3.2) as as a double integral with variables of integration u and v. a (x, y) a(u, v) du dy ru Хry dy du l|ru Xr, || f (x (u, v),y(u, v),z (u, v)) 1(xu, Wsx,y,z) Mos u.v.gou,» @ +()*+1 li ser(u, v),y(u, v),z (u, v) Date f (u, v,...
5. Suppose σ is a parametric surface with vector equation r(14. u) x (u, u)i + y(u, u)j + z(u, v)k If σ has no self-intersections and σ 1s smooth on a region R in the uu-plane, then the surface area of ơ is given by
5. Suppose σ is a parametric surface with vector equation r(14. u) x (u, u)i + y(u, u)j + z(u, v)k If σ has no self-intersections and σ 1s smooth on a region R...
(1) Let P denote the solid bounded by the surface of the hemisphere z -Vl-r-y? and the cone2y2 and let n denote an outwardly directed unit normal vector. Define the vector field F(x, y, z) = yi + zVJ + 21k. (a) Evaluate the surface integral F n dS directly without using Gauss' Divergence Theorem. (b) Evaluate the triple integral Ш div(F) dV directly without using Gauss' Diver- gence Theorem Note: You should obtain the same answer in (a) and...
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