### Create a StudySoup account

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

Already have a StudySoup account? Login here

# MULTIDIMENSIONAL CAL MATH 2057

LSU

GPA 3.72

### View Full Document

## 29

## 0

## Popular in Course

## Popular in Mathematics (M)

This 6 page Study Guide was uploaded by Madison Gottlieb Sr. on Tuesday October 13, 2015. The Study Guide belongs to MATH 2057 at Louisiana State University taught by C. Bremer in Fall. Since its upload, it has received 29 views. For similar materials see /class/222644/math-2057-louisiana-state-university in Mathematics (M) at Louisiana State University.

## Reviews for MULTIDIMENSIONAL CAL

### 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: 10/13/15

klidternl 1 Stiuly Cuide Math 2057 Section 7 22610 The Midterm will cover sections 115 141141 No books or calculators are allowed There will be four truefalse questions and six long form questions I will hold a study session 430530 PM on Thursday 115 In this section we covered the standard equations for ellipses and hyperbolas These were important for sketching the level curves of a function so it is likely that this material will appear in that context Given a quadratic equation you should be able to determine if it is a hyperbola or an ellipse and then give a reasonable sketch of that curve This section was mainly concerned with graphingl functions in several variables and nding the domains of functions You should know how to sketch the vertical and horizontal traces of a function and be able to make a rough estimate of the shape of the graph using the traces You should also be able to estimate things like average rate of change using the contour map of a function The de nition of limits and continuity in several variables will most likely be covered in the truefalse problems Given a function you should be able to use theorems l and 2 to deter mine where the function is continuous One of the tricks we used to show that a function is not continuous is to show that taking the limit from two different directions gives you different answers This section covered partial derivatives You should be aware of the relationship between partial derivatives and average rate of change at a point see section 141 Some of the trickier problems in this section asked you to show that a certain function satis ed a partial differential equation however these simply involve calculating the right and left side of an equation and showing that they are equali Clairaut7s theorem says that under reasonable conditions fgcy fygci l illustrated this rule in class using the contour map of a function A function is differentiable if it is locally lineari7 In particular this means that the tangent plane at a point P is a good approximation for the function near P although it gets worse as you get further from P You should be able to use the equation of the tangent plane to estimate the value of the function as well as estimate Af as a function of AI and Ayi Directional derivatives and chain rule part 1 Notice that the gradient has many different inter pretations Speci cally it appears in the chain rule directional derivatives and it is the normal vector to a level curve or surface The gradient points in the direction of greatest increase7 and the magnitude of the gradient is the maximal directional derivativei Given the level curves of a function you should be able to sketch the gradient at a point as well as determine if the directional derivative is positive or negative in a given directioni Pay special attention to theorem 4 and be sure that you understand the statement of the theoremi Chain Rule part 2 This gives you the chain rule for a slightly more involved change of variables You should be able to apply the general version of the chain rule as in the webwork assignment to calculate the partial derivatives of a composite function In this section we found critical points of functions in two variables using the gradient and then determined whether or not they are local maxima or minima using the second derivative testi Remember that a function with a closed bounded domain always has a global maximum and rninirnurni In order to nd the extrerna7 nd critical points in the interior of the dornain7 and then test the boundary of the domain for critical points Theorem 3 is good truefalse rnateriali FHnalStudy Chnde Math 2057 Section 2 5610 The nal will occur Wednesday May 12 at 1000 AMi You will have 2 hours to complete the exami The length will be about a midterm and a half The nal will be cumulative but there is not enough room for everything There will be a study session Monday 430530 on the second oor of Locketti The most important topics will be covered gradients chain rule optimization second derivative test double integrals over general regions and the change of variables formulas polar cylindrical and spherical coordinates as well as the general case should appear in some ormi Special emphasis will be placed on surface and line integrals conservative vector elds Greenls theorem and Stokes7 Theorem since these topics have not yet been covered on an exami There WILL be a problem requiring you to complete the square For sample problems see the previous examswebworkshomeworks and the chapter reviews from the book 1 have included outlines of each section to help you get starte i 1115 In this section we covered the standard equations for ellipses and hyperbolas These were important for sketching the level curves of a function so it is likely that this material will appear in that context Given a quadratic equation you should be able to determine if it is a hyperbola or an ellipse and then give a reasonable sketch of that curve 141 This section was mainly concerned with graphingl functions in several variables and nding the domains of functions You should know how to sketch the vertical and horizontal traces of a function and be able to make a rough estimate of the shape of the graph using the traces You should also be able to estimate things like average rate of change using the contour map of a function 142 The de nition of limits and continuity in several variables will most likely be covered in the truefalse problems Given a function you should be able to use theorems 1 and 2 to deter mine where the function is continuous One of the tricks we used to show that a function is not continuous is to show that taking the limit from two different directions gives you different answers 1413 This section covered partial derivatives You should be aware of the relationship between partial derivatives and average rate of change at a point see section 1411 Some of the trickier problems in this section asked you to show that a certain function satis ed a partial differential equation however these simply involve calculating the right and left side of an equation and showing that they are equali Clairaut7s theorem says that under reasonable conditions fgcy fygci l illustrated this rule in class using the contour map of a function 144 A function is differentiable if it is locally linear7 In particular this means that the tangent plane at a point P is a good approximation for the function near P although it gets worse as you get further from P You should be able to use the equation of the tangent plane to estimate the value of the function as well as estimate Af as a function of AI and Ayi 1415 Directional derivatives and chain rule part 1 Notice that the gradient has many different inter pretations Speci cally it appears in the chain rule directional derivatives and it is the normal vector to a level curve or surface The gradient points in the direction of greatest increasel and the magnitude of the gradient is the maximal directional derivativei Given the level curves of a function you should be able to sketch the gradient at a point as well as determine if the directional derivative is positive or negative in a given directioni Pay special attention to theorem 4 and be sure that you understand the statement of the theoremi Chain Rule part 2 This gives you the chain rule for a slightly more involved change of variables You should be able to apply the general version of the chain rule as in the webwork assignment to calculate the partial derivatives of a composite function In this section we found critical points of functions in two variables using the gradient and then determined whether they were local maxima or minima using the second derivative testi Theorem 3 is good TrueFalse materiali More maximization but with a constraint Given a constraint and an objective function you should be able to set up the Lagrange multipliers problem to nd the critical values You should also be able to use LaGrange multipliers on the boundary of a closed bounded domain in addition to nding critical points inside the domain nd global maxima and minimal Riemann sumsi You should realize that integrals are actually de ned by a limit of Riemann sums but calculated by Fubini s theoremi You should also be able to calculate simple Riemann sums as in the chapter Given a domain of integration you should be able to sketch the domain and set up the corresponding iterated integrali You need to be careful about changing the order of integrationiit is highly recommended that you sketch the domain rst before nding the new bounds of integration You should also be aware of things like the average value of functions the mean value theorem etc More of the same in three dimensions Here it becomes much more dif cult to visualize the domain of integration This takes some practice Know how to nd the projection of a domain in three dimensions onto the xy planer It is likely that any problem taken from this section will involve polar coordinates only You should know some of the basic examples of polar regions eg 7 lt ncost9 and be able to translate between Cartesian coordinates and polar coordinates Don7t forget that dzdy Td39rdt You should know the change of variables formula for spherical coordinates and be able to translate between spherical and rectangular coordinatesi Note that spherical coordinates and cylindrical coordinates are very useful when parameterizing surfaces This section involves the change of variables formula in two variables You will be expected to be able to do the following calculate the image of a domain under a transformation nd a linear transformation that takes the unit square triangle to a parallelogram respi triangle and calculate the Jacobian of a linear transformation Finally given a transformation T S A R you should be able to use the change of variables theorem to rewrite an integral ffR fz ydzdy as an integral of the form 5 vyu vdudvi The concept of a one to one map will likely be covered in the truefalse section You should be able to reasonably sketch a vector eld on a grid lf is a vector eld the two important things to consider are the magnitude at a point and the direction of You should also be able to determine whether a vector eld is a gradient eld there is some overlap with section 163 Remember that when a V415 the direction of R is normal to the level curves of The cross partials7 test theorem 1 is another important test of whether is a gradient eld In this section we saw how to integrate functions and vector elds over parameterized curvesi There is a simple formula theorem 1 for calculating a scalar line integrali Integrating a vector eld is not bad either however you need to remember that the orientation of the curve mattersi You should have some intuition for what integrating vector elds means7 in particular if I give you a picture of a vector eld and an oriented curve you should be able to say whether the integral is positive or negative see gure 7 More generally you should be aware that integrating a vector eld is a way of calculating workil Again theorem 3 is good truefalse materiali Conservative vector elds satisfy many nice properties You should recognize the main theorems of this section and understand what they mean Theorem 3 tells you that every conservative vector eld is a gradient eldi Theorem 1 is important because it allows you to integrate conservative vector elds just by knowing the potential function which is useful especially if the integral itself is dif cult or impossible Finally you need to use theorem 4 carefully We saw that the simply connected7 condition is essential for this theorem to wor You should know some of the parameterizations of standard surfaces spheres ellipses helicoids and graphs of functions Given a parameterization you need to be able to nd the formula for the normal vector This involves a cross product Then theorem 1 gives you a way to integrate functions over a parameterized surface using the magnitude of the normal vector uvi If you are having trouble nding parameterizations see the remark in section 154 Surface integrals over vector elds While surface integrals of functions do not depend on orienta tion surface integrals of vector elds doi Therefore when you nd the normal vector you need to make sure that it has the correct orientation If you have the wrong orientation you just need to multiply FL by 71 Again you should memorize the parameterizations of the standard surfaces It will be especially helpful to practice with graphs of functions To see the comparison between line and surface integrals of functions and vector elds consider the following handy chart Let C be a curve parameterized by Ct a S t S b and let S be a surface parameterized by uw uv E D where D is a domain in R l Functions fz y 2 l Vector Fields Fz y 2 Curves C C fds ff fctHc tHdt c F ds fFctgt alttgtgtdt Surfaces S 5 de j ffD f uvll uvlldvdu ffSF d5 ffD F u 11 uvdvdu Greenls Theoremi There are some good exercises involving Greenls theorem on the webwork assignment and it is likely that the exam question will be similar Remember that you need to orient your curve counterclockwise for Green s theorem to give the right answer or more generally the interior of the domain should be to the left of the direction of travel Most problems on Greenls theorem involve either nding the area of a domain by integrating 12 g zgt around its boundary OR involve the calculation of a line integral by changing it to a double integral It always helps to look at a sketch Stokels theorem is a generalization of Green s theorem for surfaces You need to be able to use the right hand rule to orient the boundary of a surface You should also memorize the formula for the curl of a vector eld Note that this version of curll gives a vector eld instead of a function As before a vector eld has curl 0 if and only if the mixed partials are equal therefore if the domain is simply connected the vector eld is a gradient eldi Finally an application of Stokels theorem requires you to change a surface integral to a line integral or vice versa and then compute that integrali k idterni 2 Stiuly Cuide ilath 2057 41609 The second midterm will cover chapter 15 i No books or calculators are allowed There will be four truefalse questions and around six long form questions There will be a study session 430530 PM in Lockett 284 Thursday 415 151 Integration in Several Variables In this section we covered Riemann sums and Fubinils theoremi Fubinils theorem allows you to convert a double integral over a rectangle into an iterated integrali Given a partition of a rectangle and a collection of sample points you should be able to calculate a Riemann sum that estimates the double integrali Note the distinction between the de nition of the integral and theorem 3 Double Integrals over General Regions You should be able to describe a vertically respi horizon tally simple region in terms of inequalities involving functions of z respi Changing the order of integration is not as simple as it is for rectangular regions so you should be familiar with this technique It is very helpful to sketch the region rst A common mistake is to leave the bounds of integration for the last variable integrated as functions they should be constantsi Finally you should be able to calculate the average value of a function on a domain and break a domain into smaller pieces to calculate the integral if necessary Triple lntegrals Triple integrals over general domains are trickyi You should be able to nd the projection of a domain in R3 onto t e z 7 y respi z 7 z y 7 2 plane and then set up an iterated integral over that domain You should be able to change the order of integration for wedgeshaped regions Again it is helpful to be able to sketch the region Polar Cylindrical and Spherical coordinates When changing to different coordinate systems there will always be a correction term in the integrand You should memorize the correction term for polar cylindrical and spherical coordinates Most of the tricky examples involve polar coordinates7you should be familiar with some standard polar regions eigi example 2 Cylindrical coordinates are very similar to polar coordinates however spherical coordinates are useful for shapes like spheres and ice cream cones7 Change of Variables Be familiar with the terminology introduced in the rst section of 155 map7 image range7 domain etc The distinction between onetoone and onto will likely be covered in the truefalse questions The only case where you will be expected to come up with a mapping on your own is in the case of a linear map erg maps taking the unit rectangle to a parallelogram or maps taking the unit circle to an ellipse However you should be able to nd the range of a general map You should also be able to calculate the Jacobian of a map and apply the change of variables formula for an integral We did not cover change of variables in three variables

### 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!"

#### "I signed up to be an Elite Notetaker with 2 of my sorority sisters this semester. We just posted our notes weekly and were each making over $600 per month. I LOVE StudySoup!"

#### "Knowing I can count on the Elite Notetaker in my class allows me to focus on what the professor is saying instead of just scribbling notes the whole time and falling behind."

#### "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.