Chapter 1. calc_tutorial_16_4_010

1.1 Problem Statement

{3,5,7,9}

pow($a,2)

$b-1

2*$a

Calculate T_r, T_θ, and n(r, θ) for the parametrized surface at the given point. Then find the equation of the tangent plane to the surface at that point.

G(r, θ) = (r cos θ, r sin θ, 1 - r²); r = 1 / $a, θ = π / 4

1.2 Step 1

Question Sequence

Question 1.1

Recall how to find the tangent vectors, T_r, T_θ, and the normal vector, n(r, θ), to a given parametrized surface, G(r, θ).

T_r =

JLEu7EXpIxt/mRKT1TX0uvdZJ9zUn4fp97H8eiW7KuvGkP0B9zBYZa+QavidDZoRwtnlKH6PxISRMsfmOiNVyHwnsw1o9fp9HHxE+8AycPOVsb+x5J0ILMJ4ULGJXCv4u2k4D+HU9rYCuFOcmSEVRw==

T_θ =

EXmwi12BxQlVBrmS2LnKVidjUgR1B0boqFIZaiPAT5ojGURqKvDwLpglyboRUqTxG+gyFX4/h1fmEJSDGWVeiKDeyqPcX3WRwAPD9nH/K9NYaw2wTBHdBQnDKLA6UQAYgXfcfQxL2tQ6iWe7iDYVSg==

n(r, θ) = T_r × T_θ

Correct.

Incorrect.

1.3 Step 2

Question Sequence

Question 1.2

Find the general expressions for T_r and T_θ given G(r, θ) = (r cos θ, r sin θ, 1 - r²)

T_r = (∂ / ∂r)(r cos θ, r sin θ, 1 - r²)

= <cos θ, sin θ, FmAhxBkQqN0=r>

T_θ = (∂ / ∂θ)(r cos θ, r sin θ, 1 - r²)

= <-r sin θ, r cos θ, 1Wh3cvJ2xF4=>

Calculate T_r and T_θ at the point where r = 1 / $a and θ = π / 4

T_r = <1/√2, 1/√2, -2/nc1ItEz0kR4=>

T_θ = <-1/(nc1ItEz0kR4=√2), 1/(nc1ItEz0kR4=√2), 0>

Correct.

Incorrect.

1.4 Step 3

Question Sequence

Question 1.3

Find the general expression for n(r, θ). Let the table below represent the matrix A

i	j	k
cos θ	sin θ	-2r
-r sin θ	r cos θ	0

Table : Matrix A

n(r, θ) = T_r × T_θ

= <cos θ, sin θ, -2r> × <-r sin θ, r cos θ, 0>

= det(A)

= r<XvVM00l89Is=r cos θ,XvVM00l89Is=r sin θ, 0VV1JcqyBrI=>

Calculate n(r, θ) at the point when r = 1 / $a and θ = π / 4

n(1 / $a, π / 4) = <√2/iSba6t70dtA=, √2/iSba6t70dtA=, 1/nc1ItEz0kR4=>

Correct.

Incorrect.

1.5 Step 4

Question Sequence

Question 1.4

We now wish to find the equation of the tangent plane to the surface G(r, θ) = (r cos θ, r sin θ, 1 - r²) at the point where r = 1 / $a and θ = π / 4.

Recall that an equation of the plane through the point P = <x₀, y₀, z₀> with normal vector n can be described as follows.

<x - x₀, y - y₀, z - z₀> · n = 0

Find the coordinates of the point on the surface G(r, θ) at the point where r = 1 / $a and θ = π / 4.

P(x, y, z) = ((1/$a)cos(π/4), (1/$a)sin(π/4), 1 - (1/$a)²)

= (1/(nc1ItEz0kR4=√2), 1/(nc1ItEz0kR4=√2), SFgqQUkJGdg=/$b)

Correct.

Incorrect.

1.6 Step 5

Question Sequence

Question 1.5

Find the equation of the tangent plane to G(r, θ) = (r cos θ, r sin θ, 1 - r²) at the point where r = 1 / $a and θ = π / 4

We will use the equation <x - x₀, y - y₀, z - z₀> · n = 0. The point on our surface is P = (1/($a√2), 1/($a√2), $c/$b) and the normal vector to the surface at that point is n = (1/$a)<√2/$a, √2/$a, 1>. It suffices to use the normal vector n = <√2/$a, √2/$a, 1> since the scalar factor of 1/$a will not affect the equation of the plane.

Write the equation of the tangent plane by computing the dot product.

<x - x₀, y - y₀, z - z₀> · n = 0

<x - (1/($a√2), y - 1/($a√2), z - ($c/$b)> · <√2/$a, √2/$a, 1> = 0

(√2/nc1ItEz0kR4=)(x - (√2/U2GIbglD1oM=)) + (√2/$a)(y - (√2/U2GIbglD1oM=)) + z - ($c/iSba6t70dtA=) = 0

Correct.

Incorrect.