multivariable calculus, vector calculus, vectors ### Problem 9On the last day of class, we described how two curves in $\mathbb{R}^2$ being **tangent to each other** corresponded to their normal vectors being parallel (and this helped explain why Lagrange multipliers work). Now we try to generalize this idea of tangency in various ways.**(a)** Draw a picture of two spheres of different sizes being tangent to each other at exactly one point $ P $ in $\mathbb{R}^3$. **Do NOT** put one sphere inside the other one.**(b)** Suppose we have two surfaces $ S_1 $ and $ S_2 $ which intersect at some point $ P = (x_0, y_0, z_0) $, and further suppose that the normal vectors $ \vec{n}_1 = \langle a_1, b_1, c_1 \rangle $ (for $ S_1 $) and $ \vec{n}_2 = \langle a_2, b_2, c_2 \rangle $ (for $ S_2 $) at $ P $ are parallel (and non-zero). Prove that the tangent planes $ T_P S_1 $ and $ T_P S_2 $ are the same by transforming the scalar equation for $ T_P S_1 $ into the scalar equation for $ T_P S_2 $. **Hint:** This should be a one-step algebraic transformation based upon using the algebraic definition of parallel vectors.### Explanation for Educational WebsiteThis problem helps illustrate the concept of tangency and how it can be extended from $\mathbb{R}^2$ (two dimensions) to $\mathbb{R}^3$ (three dimensions). By examining how normal vectors of surfaces behave, we can understand why and how tangency works in higher dimensions.**Part (a)** asks students to visualize and draw two spheres that touch at exactly one point. This part emphasizes the spatial understanding of tangent surfaces. **Part (b)** is a more algebraic approach, asking students to prove the equality of tangent planes using parallel normal vectors. Parallel normal vectors imply that the corresponding tangent planes must coincide. The hint suggests this can be done in a straightforward algebraic step, encouraging the use of fundamental vector operations. **Tangent Surfaces and Linearization Functions**Based on what we have so far, we will say two surfaces $S_1$ and $S_2$ are **tangent to each other** at an intersection point $P$ when they have parallel normal vectors at $P$, which is the same as saying that they have the same tangent plane at $P$.(c) If our tangent surfaces are graphs $S_1 = (\text{graph of } f_1(x,y))$ and $S_2 = (\text{graph of } f_2(x,y))$, what can you say about the linearization functions $\mathcal{L}_P f_1(x,y)$ and $\mathcal{L}_P f_2(x,y)$? Provide a one-to-two sentence explanation for your claim.

a) The objective is to draw two spheres of different sizes tangent to each other at exactly one…

Answered: (a) exactly one point P in R³. Do NOT…

(a) exactly one point P in R³. Do NOT put one sphere inside the other one. Draw a picture of two spheres of different sizes being tangent to each other at

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Arc Length

Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length.

Line Integral

A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.

Triple Integral

Examples:

Question

multivariable calculus, vector calculus, vectors

### Problem 9

On the last day of class, we described how two curves in $\mathbb{R}^2$ being **tangent to each other** corresponded to their normal vectors being parallel (and this helped explain why Lagrange multipliers work). Now we try to generalize this idea of tangency in various ways.

**(a)** Draw a picture of two spheres of different sizes being tangent to each other at exactly one point $ P $ in $\mathbb{R}^3$. **Do NOT** put one sphere inside the other one.

**(b)** Suppose we have two surfaces $ S_1 $ and $ S_2 $ which intersect at some point $ P = (x_0, y_0, z_0) $, and further suppose that the normal vectors $ \vec{n}_1 = \langle a_1, b_1, c_1 \rangle $ (for $ S_1 $) and $ \vec{n}_2 = \langle a_2, b_2, c_2 \rangle $ (for $ S_2 $) at $ P $ are parallel (and non-zero). Prove that the tangent planes $ T_P S_1 $ and $ T_P S_2 $ are the same by transforming the scalar equation for $ T_P S_1 $ into the scalar equation for $ T_P S_2 $. **Hint:** This should be a one-step algebraic transformation based upon using the algebraic definition of parallel vectors.

### Explanation for Educational Website

This problem helps illustrate the concept of tangency and how it can be extended from $\mathbb{R}^2$ (two dimensions) to $\mathbb{R}^3$ (three dimensions). By examining how normal vectors of surfaces behave, we can understand why and how tangency works in higher dimensions.

**Part (a)** asks students to visualize and draw two spheres that touch at exactly one point. This part emphasizes the spatial understanding of tangent surfaces.

**Part (b)** is a more algebraic approach, asking students to prove the equality of tangent planes using parallel normal vectors. Parallel normal vectors imply that the corresponding tangent planes must coincide. The hint suggests this can be done in a straightforward algebraic step, encouraging the use of fundamental vector operations.

**Tangent Surfaces and Linearization Functions**

Based on what we have so far, we will say two surfaces $S_1$ and $S_2$ are **tangent to each other** at an intersection point $P$ when they have parallel normal vectors at $P$, which is the same as saying that they have the same tangent plane at $P$.

(c) If our tangent surfaces are graphs $S_1 = (\text{graph of } f_1(x,y))$ and $S_2 = (\text{graph of } f_2(x,y))$, what can you say about the linearization functions $\mathcal{L}_P f_1(x,y)$ and $\mathcal{L}_P f_2(x,y)$? Provide a one-to-two sentence explanation for your claim.

Calculus that deals with differentiation and integration of the vector field in three-dimensional Euclidean space. It deals with quantities that have both magnitude and direction.

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