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3. If T2 = r3 cos(0) sin(d) and v2 = sin(0) cos(O)f + r sin(0)θ +...
(2) Let x-r cos θ, y-r sin θ represent the polar coordinates function f(r, θ) : R. R2, Compute f,...
(2) Let x-r cos θ, y-r sin θ represent the polar coordinates function f(r, θ) : R. R2, Compute f, (r$) and f, ( ompute * T
(2) Let x-r cos θ, y-r sin θ represent the polar coordinates function f(r, θ) : R. R2, Compute f, (r$) and f, ( ompute * T
5) Let Φ : R2-ל -(rcos(0), r sin(θ)), 0-r-R, 0-θ disk of radius R centered at (0,0)). Compute J dx Λ dy. R2 given by Φ(r, θ) -2n (this is a 5) Let Φ : R2-ל -(rcos(0), r sin(θ)), 0-r-R, 0-θ disk...
5) Let Φ : R2-ל -(rcos(0), r sin(θ)), 0-r-R, 0-θ disk of radius R centered at (0,0)). Compute J dx Λ dy. R2 given by Φ(r, θ) -2n (this is a
5) Let Φ : R2-ל -(rcos(0), r sin(θ)), 0-r-R, 0-θ disk of radius R centered at (0,0)). Compute J dx Λ dy. R2 given by Φ(r, θ) -2n (this is a
16, Let x: U R2-, R, where x(8, φ) (sin θ cos φ, sin θ sin φ, cos θ), be a parametrization of the...
16, Let x: U R2-, R, where x(8, φ) (sin θ cos φ, sin θ sin φ, cos θ), be a parametrization of the unit sphere S2. Let and show that a new parametrization of the coordinate neighborhood x(U) = V can be given by y(u, (sech u cos e, sech u sin e, tanh u Prove that in the parametrization y the coefficients of the first fundamental form are Thus, y-1: V : S2 → R2 is a conformal...
An ellipsoid, S, is parameterised by r = a sin θ cos φi + a sin...
An ellipsoid, S, is parameterised by r = a sin θ cos
φi + a sin θ sin
φj + b cos θk 0
≤ θ ≤ π 0 ≤ φ ≤ 2π
i. Find the surface element dS, such that
dS points OUT of the ellipsoid.
ii. Hence determine the following surface integral over the
ellipsoid:
//rds JJs
Consider the following surface parametrization. x-5 cos(8) sin(φ), y-3 sin(θ) sin(p), z-cos(p) Find an expression for a unit vector, n, normal to the surface at the image of a point (u, v) for θ in [...
Consider the following surface parametrization. x-5 cos(8) sin(φ), y-3 sin(θ) sin(p), z-cos(p) Find an expression for a unit vector, n, normal to the surface at the image of a point (u, v) for θ in [0, 2T] and φ in [0, π] -3 cos(θ) sin(φ), 5 sin(θ) sin(φ),-15 cos(q) 16 sin2(0) sin2(p)216 cos2(p)9 3 cos(9) sin(9),-5 sin(θ) sin(9), 15 cos(q) 16 sin2(0) sin2(p)216 cos2(p)9 v 16 sin2(0) sin2@c 216 cos2@t9(3 cos(θ) sin(φ), 5 sin(θ) sin(φ) , 15 cos(q) 216 cos(φ)...
(3) For the following velocity fields F on R3, find the flow along the given curve. r(t) = (t, t2, 1) F=(-4xy, 83, 2) with 0 2 t 1l F=(z-z, 0,2) r(t)-(cost, 0, sin t) with 0 t π F = (-y,2, 2) with r(...
(3) For the following velocity fields F on R3, find the flow along the given curve. r(t) = (t, t2, 1) F=(-4xy, 83, 2) with 0 2 t 1l F=(z-z, 0,2) r(t)-(cost, 0, sin t) with 0 t π F = (-y,2, 2) with r(t) = (-2 cost, 2 sin t, 2t) 0 < t < 2π
(3) For the following velocity fields F on R3, find the flow along the given curve. r(t) = (t, t2, 1) F=(-4xy, 83,...
3. A general surface of revolution is r(u, θ)-(f( u') cos θ , f(u) sin θ,...
3. A general surface of revolution is r(u, θ)-(f( u') cos θ , f(u) sin θ, υ), θΕ[0, 27), where f(u) is a positive function. For the following choices of f(u), find the principal, Gaus- sian, and mean curvatures at arbitrary (u, θ), and classify each point on the surface as elliptic, hyperbolic, parabolic, or planar. (a) f(u)u, u E [0, 00) (b) f(uV1 - u2,u e[-1,1].
Let g: R3 R3 be the cylindrical coordinate transformation g(r, 0, 2) (r cos e,r sin...
Let g: R3 R3 be the cylindrical coordinate transformation g(r, 0, 2) (r cos e,r sin 0, 2). Which of the following is equal to det(Dg(r, 0, 2))? Ο η r cos(0) O-rsin(O)
3. Let z,-34J4 and z2-10e'. Find the results of: zit 22,21 * 22,21/22, ะ?, and å . Find the Laplace transforms of the following functions: note that t2 0 1). f(t) =e®.4tcos 12t. f) sin(t 3),...
3. Let z,-34J4 and z2-10e'. Find the results of: zit 22,21 * 22,21/22, ะ?, and å . Find the Laplace transforms of the following functions: note that t2 0 1). f(t) =e®.4tcos 12t. f) sin(t 3), f(t) =cos2wtcos3wt. Hint: You may want to use the following identities: cos(o + β)-cosoosß-sino sind, sin θ = , 2 2j
3. Let z,-34J4 and z2-10e'. Find the results of: zit 22,21 * 22,21/22, ะ?, and å . Find the Laplace transforms of the...
Consider the following matrix, As[ cos θ sin θ -L-sin θ cos θ J, for some...
Consider the following matrix, As[ cos θ sin θ -L-sin θ cos θ J, for some θ E (-π, π] (a) What is determinant of A? (b) Perform an eigen-decomposition of A (c) What does this matrix do to a vector in R2.
(2) Let x-r cos θ, y-r sin θ represent the polar coordinates function f(r, θ) : R. R2, Compute f, (r$) and f, ( ompute * T
(2) Let x-r cos θ, y-r sin θ represent the polar coordinates function f(r, θ) : R. R2, Compute f, (r$) and f, ( ompute * T
5) Let Φ : R2-ל -(rcos(0), r sin(θ)), 0-r-R, 0-θ disk of radius R centered at (0,0)). Compute J dx Λ dy. R2 given by Φ(r, θ) -2n (this is a
5) Let Φ : R2-ל -(rcos(0), r sin(θ)), 0-r-R, 0-θ disk of radius R centered at (0,0)). Compute J dx Λ dy. R2 given by Φ(r, θ) -2n (this is a
16, Let x: U R2-, R, where x(8, φ) (sin θ cos φ, sin θ sin φ, cos θ), be a parametrization of the unit sphere S2. Let and show that a new parametrization of the coordinate neighborhood x(U) = V can be given by y(u, (sech u cos e, sech u sin e, tanh u Prove that in the parametrization y the coefficients of the first fundamental form are Thus, y-1: V : S2 → R2 is a conformal...
An ellipsoid, S, is parameterised by r = a sin θ cos
φi + a sin θ sin
φj + b cos θk 0
≤ θ ≤ π 0 ≤ φ ≤ 2π
i. Find the surface element dS, such that
dS points OUT of the ellipsoid.
ii. Hence determine the following surface integral over the
ellipsoid:
//rds JJs
Consider the following surface parametrization. x-5 cos(8) sin(φ), y-3 sin(θ) sin(p), z-cos(p) Find an expression for a unit vector, n, normal to the surface at the image of a point (u, v) for θ in [0, 2T] and φ in [0, π] -3 cos(θ) sin(φ), 5 sin(θ) sin(φ),-15 cos(q) 16 sin2(0) sin2(p)216 cos2(p)9 3 cos(9) sin(9),-5 sin(θ) sin(9), 15 cos(q) 16 sin2(0) sin2(p)216 cos2(p)9 v 16 sin2(0) sin2@c 216 cos2@t9(3 cos(θ) sin(φ), 5 sin(θ) sin(φ) , 15 cos(q) 216 cos(φ)...
(3) For the following velocity fields F on R3, find the flow along the given curve. r(t) = (t, t2, 1) F=(-4xy, 83, 2) with 0 2 t 1l F=(z-z, 0,2) r(t)-(cost, 0, sin t) with 0 t π F = (-y,2, 2) with r(t) = (-2 cost, 2 sin t, 2t) 0 < t < 2π
(3) For the following velocity fields F on R3, find the flow along the given curve. r(t) = (t, t2, 1) F=(-4xy, 83,...
3. A general surface of revolution is r(u, θ)-(f( u') cos θ , f(u) sin θ, υ), θΕ[0, 27), where f(u) is a positive function. For the following choices of f(u), find the principal, Gaus- sian, and mean curvatures at arbitrary (u, θ), and classify each point on the surface as elliptic, hyperbolic, parabolic, or planar. (a) f(u)u, u E [0, 00) (b) f(uV1 - u2,u e[-1,1].
Let g: R3 R3 be the cylindrical coordinate transformation g(r, 0, 2) (r cos e,r sin 0, 2). Which of the following is equal to det(Dg(r, 0, 2))? Ο η r cos(0) O-rsin(O)
3. Let z,-34J4 and z2-10e'. Find the results of: zit 22,21 * 22,21/22, ะ?, and å . Find the Laplace transforms of the following functions: note that t2 0 1). f(t) =e®.4tcos 12t. f) sin(t 3), f(t) =cos2wtcos3wt. Hint: You may want to use the following identities: cos(o + β)-cosoosß-sino sind, sin θ = , 2 2j
3. Let z,-34J4 and z2-10e'. Find the results of: zit 22,21 * 22,21/22, ะ?, and å . Find the Laplace transforms of the...
Consider the following matrix, As[ cos θ sin θ -L-sin θ cos θ J, for some θ E (-π, π] (a) What is determinant of A? (b) Perform an eigen-decomposition of A (c) What does this matrix do to a vector in R2.
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