### Create a StudySoup account

#### Be part of our community, it's free to join!

Already have a StudySoup account? Login here

# MATH 2010 - Multivariable Calculus and Matrix Algebra (Herron) - Week 2 Notes (S16) MATH 2010

RPI

GPA 3.39

### View Full Document

## About this Document

## 138

## 2

1 review

## Popular in Multivariable calculus and matrix algebra

## Popular in Mathematics (M)

This 5 page Class Notes was uploaded by creask on Sunday January 31, 2016. The Class Notes belongs to MATH 2010 at Rensselaer Polytechnic Institute taught by Isom Herron in Spring 2016. Since its upload, it has received 138 views. For similar materials see Multivariable calculus and matrix algebra in Mathematics (M) at Rensselaer Polytechnic Institute.

## Popular in Mathematics (M)

## Reviews for MATH 2010 - Multivariable Calculus and Matrix Algebra (Herron) - Week 2 Notes (S16)

These are great! I definitely recommend anyone to follow this notetaker

-*Jasmin Blick*

### What is Karma?

#### Karma is the currency of StudySoup.

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 01/31/16

MATH 2010 – Multivariable Calculus & Matrix Algebra Professor Herron – Rensselaer Polytechnic Institute Week 2 (2/1/16 – 2/5/16) Important: These notes are in no way intended to replace attendance in lecture. For best results in this course, it is imperative that you attend lecture and take your own detailed notes. Please keep in mind that these notes are written specifically with Professor Herron’s sections in mind, and no one else’s. There were no important pictures or diagrams in lecture this week, so I will not be placing images in this set of notes. Gradients - Two kinds of functions Real value functions o ???? = ???? ????,???? = ???? ???? ( ⃑ Vector value functions o ???? ⃑ = ???? ???? ; ????⃑ = ???? ,???? 〉and ????⃑ = ???? ,???? 〉 1 2 1 2 - A gradient is a special type of vector value function, defined by partial derivatives If ???? is a function of two variables, ???? and ????, then the gradient of ???? is the vector ???????? ???????? function ∇???? defined by: ∇???? ????,???? = ???? ????,????????,???? ????,???? ???? ( )〉 = ????????????̂ +???????? ????̂ ???????? ???????? ????????̂ In three dimensions: ∇???? ????,????,???? = ???? ,???? ???????? ???? ????〉 = ????????????̂ +???????? ????̂ +???????????? Example 1 - ???? = ???? + ???? ; find the gradient and the value of ????⃑. ∇???? ????,???? = 2????????̂ + 2????????̂ = 2???? ⃑ ⃑ = ????????̂ + ????????̂ ‖???? = √ ???? + ???? (the level curves are circles) iClicker - A unit vector in the direction of vector ????̂ = ????,????,???? has formulae: 1 ‖????‖⃑ 1 √???? +???? +???? 2⃑ 1 ‖????‖〈????,????,???? 〉 Applications of Gradients: 1 - Chain rule for paths Suppose a particle is moving on a surface ???? = ????(????,????). Its position is (???? ???? ,???? ???? ,???? ???? ). Find its velocity vector. ???? ???? = ???? ????(???? ???? ,???? ???? ) = ???????? ????????+ ???????? ???????? ???????? ???????? ???????? ???????? ???????? Suppose that ???? = ????(????,????) is a differentiable function of ???? and ????, where ???? = ????(????) and ???? = ℎ(????) are both differentiable forms of ????. Then ???? is a differentiable function ???????? ???????? ???????? ???????? ???????? of ???? and???????? = ???????? ???????? + ???????? ????????. Example 2 2 2 ???????? - ???? = ????((???????????????? ???? ) ,(???????????????? ???? ) ). Find ????????. ???????? ???????? ???????? ( ) ( ) ???????? ( ) ???????? = 2????; ???????? = 2????; ???????? = cos ???? − ???????????????? ???? ; ????????= sin ???? + ????????????????(????) ???????? ???????? ???????? ???????? ???????? ????????+ ???????? ????????= 2(???????????????? ???? )(cos ???? − ???????????????? ???? ) + 2(???????????????? ???? )(sin ???? + ???????????????? ???? ) ( ) = 2???????????????? ???? − 2???? cos ???? sin ???? + 2???????????????? ???? + 2???? cos ???? sin(????) ( ) 2 2 a) Recall that ???????????? ???? + ???????????? ???? = 1) = 2????(???????????? ???? + ???????????? ???? ) →( ) ????????= ???????? ???????? Applications of Gradients: 2 - Directional Derivatives ???????? ????(????,????) is the directional derivative ???????? ???? ???? = ???????? ???? ????,????( ) ???? ????+????ℎ,????+ℎ???? ????(????,????) lim , where (ℎ,????) are small differences ????→0 ???? The idea is to use the chain rule for paths when the path is a line segment joining (???? + ℎ,???? + ????) to (????,????) ???????? ???? ???? = ???????? ∙ ????, where ???? ̂ is the unit vector ⃑ ???? = ???? + ????ℎ,???? + ???????? → ????̂〉 ⃑ = ℎ,???? = ???? ̂ ???? ????+????ℎ,????+????????,????+???????? −????(????,????,???? In three variables: ???? olim∇???? ∙ ????⃑ = ????̂ ∙ ∇????, where ????̂ = ℎ,????,???? 〉 ????→0 Example 3 - Suppose temperature ???? = (????,????,????) at a point is given in degrees by: ???? = 50 + ????????????. Find the rate of change of the temperature with respect to distance at (3,4,1) in 〈 〉 the direction of a)⃑ = 1,2,2 and b) ???? ⃑ in the direction of the origin. a) ???????? ???? = ∇???? ∙ ???? ̂; ̂ = ????⃑⃑= 〈1,2,2 = 〈1,2,2= 1,2,2 ‖????⃑ √1 +2 +22 √9 3 o ∇???? = ???? ,???? ???????? ???? ????????= ???????????? ???? = ????????;???????? = ???????? ???? 〈1,2,2 4+6+24 o ∇???? 3,4,1 = 4,3,12 ;???????? ???? 3,4,1 = 4,3,12 ∙〈 〉 3 = 3 ???????? ???????? ???? ????,????,???? = ) ???? − 3,4,1 −1 −???? b) ????̂ = √3 +4 +1 2= √26 3,4,1 → ???????? ???? ????,????,???? = ????,????,???????? ∙ 〉 √???????? 〈????,????,???? 〉 Find the maximum directional derivative ???????? ???? at the point (3,4,1) and the direction ⃑ in which the maximum occurs. o To maximize the directional derivative: ???????? ???? = ∇???? ∙ ????̂ = ∇???? ‖‖ ̂ cos(????) ???? is the angle between ∇???? and ????̂ (???? = 0 → cos ???? = 1) o The max. value of ???????? ???? = ∇???? , when ???? ̂ has the same direction as ∇????. o Thus the min. occurs where ???? = ???? (the direction opposite of ∇????) iClicker - Find the slope of ???? = ???? ????,???? = ???? + ???? at (1,2) in the direction of 2,1 .〉 〈 〉 ∇???? 1,2 = 2,4 ; 2,4 ∙ 〉 2,1 = 4+4 → ???? √2 +12 √5 √ ???? Implicit Gradients - For any curve ⃑(????) on a surface ????(???? ???? ,???? ???? ,???? ???? ) = ????. ???? - Hence, ???????? ????(???? ???? ,???? ???? ,???? ???? ) = 0. - ∇???? ∙ ⃑ ???? = 0; ∇???? = ???? ,???? ,???? 〉 ???? ???? ???? - ∇???? is the direction of the normal vector ???? to the surface This is true for any curve. Example 4 - ???? + ???? + ???? = 3 at the point (1,−1,1). Find the tangent plane. 2 2 2 ???? = ???? + ???? + ???? ; ∇???? = 2????,2????,2???? 〉 ∇???? 1,−1,1 = 2,−2,2 = ???? 〉 ⃑⃑⃑ o The equation is: ???? ∙ ????⃑ − ????0 )= 0 o 2 ???? − 1 − 2 ???? + 1 + 2 ???? − 1 = 0 ) o ???? − 1 − ???? − 1 + ???? − 1 →???? − ???? + ???? = ???? The Chain Rule (Case 2) - Suppose ???? = ????(????,????) is a differentiable function of ???? and ????, where ???? = ????(????,????) and ???? = ℎ(????,????) are differentiable functions of ???? and ????. ???????? ???????? ???????? ???????? ???????? ???????? ???????? ???????? ???????? ???????? - Then: ???????? = ???????? ???????? + ???????? ???????? and ????????= ???????? ????????+ ???????? ???????? Applications of Gradients: 3 - Polar Coordinates ???? = ???????????????? ???? ,???? = ????????????????(????) Suppose ???? is a function ???? = ????(????,????) represented as a function of (????,????) ???????? ???? ???????? ???? ???????? = ????????(???????????????? ???? ) = cos ???? ;) ???????? = ???????? (???????????????? ???? ) = sin ???? ) ???????? ???????? ???????? ???????? ???????? ???????? = ???????? ????????+ ???????? ???????? = ???? ????−???????????????? ???? ) + ???? (???????????????? ???? ) ) ???????? = ???? ????os ????( )) + ????????(sin ???? ) ???????? 2 2 ( ????????) + ( 1 ????????) = ???? ???????????? ???? + 2???? ???? cos ???? sin ???? + ???? ???????????? ???? + 2 2( ) ???????? ???? ???????? ???? ???? ???? ???? ???? ???????????? ???? − 2???? ???? cos ???? sin ???? + ???? ???????????? ???? = ???? + ????( ) 2 2= ∇???? ‖2 ???? ???? ???? ???? ???? ???? Solving for ????????,????????in terms of ????????,???? ???? 1 o ???? ???? cos ???? ???? − s????n ???????????? ( ) ???? ( ) 1 ( ) o ???? ???? sin ???? ???? + c????s ???????????? ???? Optimization - When the tangent plane to surface ???? = ????(????,????) is horizontal (flat), there’s a chance that the graph has a local extreme value at a point. - However, it may be neither a maximum nor a minimum, but a saddle point. - When ∇???? = (0,0) at a point, this point is a critical point. Example 1 - ???? ????,???? = 2???? + ???? − ???????? − 7????. Find the local extreme. ∇???? = 4???? − ???? ????̂ + 2???? − ???? − 7 ????̂ = 0,0〈 〉 Solve (system of two equations): o 4???? − ???? = 0 o 2???? − ???? − 7 = 0 o 2 4???? − ???? + 2???? − ???? − 7 = 0 o 8???? − 2???? + 2???? − ???? − 7 = 0 → 7???? = 7 → ???? = ???? o Subbing back in: 4 1 − ???? = 0 → ???? = ???? Compare values of ???? at 1,4 with those at nearby points (1 + ℎ,4 + ????): o ???? 1,4 = 2 + 16 − 4 − 28 = −14 2 2 o ???? 1 + ℎ, 4 + ???? = 2 1 + ℎ ) + 4 + ???? ) − 1 + ℎ 4 + ????2− 7(4 + ????) o ???? 1 + ℎ,4 + ???? − ???? 1,4 = 2ℎ + ???? − ℎ???? = (???? − ) + ℎ ≥ 0 ℎ 7 2 2 4 Thus, ????(????,????) is a minimum. Example 2 2 - ???? ????,???? = ???? − ???????? + 2???? + ???? + 1. Find the local extreme. ∇???? = 2 − ???? ????̂ + 2???? − ???? + 1 ????̂ = 0,0〈 〉 o ???? = 2,???? = 5 ???? 5,2 = 4 − 10 + 10 + 2 + 1 = 7 2 ???? 5 + ℎ,???? + 2 = 4 + 4???? + ???? − ℎ???? 2 2ℎ − 5???? − 10 + 10 + 2ℎ + ???? + 2 + 1 Subtracting the two equations: ???? − ℎ???? + 7 − 7 → ℎ = ???? Thus, ????(????,????) is a saddle point. Theorem nd - If ???? ????,???? has continuous 2 partial derivatives in a notion of a critical point (????,????) and if the number ???? (the discriminant) is defined by: 2 ???? = ???? ????????(????,???? ∙ ???? ????????(????,???? − [???? ????????(????,???? ] , then ????,???? is a: a) Maximum if ???? > 0 and ???? ????????(????,???? < 0 b) Minimum if ???? > 0 and ???? ????????(????,???? > 0 c) Saddle point if ???? < 0 Example 3 - ???? = 2???? + ???? − ???????? − 7????. Find the local extreme. ???? = 4???? − ????; ???? = 2???? − ???? − 7 ???? ???? ???????????? 4;???? ???????? = −1;???? ???????? = 2 ???? = 4 ∙ 2 − −1 )2= 7 → minimum Example 4 - ???? = ???? − ???????? + ???? + 1 ???? = −????; ???? = 2???? − ???? + 1 ???? ???? ???????????? 0; ???? ???????? = −1; ???? ???????? = 2 ???? = 0 ∙ 2 − −1 )2= −1 → saddle point iClicker - ???? ????,???? = ???? − 2???? + ???? − 1. Find ∇???? = 0,0 . 〈 〉 ∇???? = 2???? − 2 ????̂ + 2???? ????̂ →(????,????) Important Note - If ???? = 0, no conclusion can be immediately drawn 2 2 ???? = 1 + 2???????? − ???? − ???? ????????= 2???? − 2????; ???? =????2???? − 2???? → ???? = ????, no critical point (ridge) - Three cases: 4 4 ???? = ???? + ????4,???? = 4;(0,0) is a local minimum ???? = − ???? + ???? ,???? = 0;(0,0) is a local maximum ???? = ???? − ???? ,???? = 0;(0,0) is a saddle point End of Document

### BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.

### You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

#### "I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

#### "When you're taking detailed notes and trying to help everyone else out in the class, it really helps you learn and understand the material...plus I made $280 on my first study guide!"

#### "I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

#### "It's a great way for students to improve their educational experience and it seemed like a product that everybody wants, so all the people participating are winning."

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.