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# MATH 2010 - Multivariable Calculus and Matrix Algebra (Herron) - Week 1 Notes (S16) MATH 2010

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This 3 page Class Notes was uploaded by creask on Saturday January 16, 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 294 views. For similar materials see Multivariable calculus and matrix algebra in Mathematics (M) at Rensselaer Polytechnic Institute.

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Date Created: 01/16/16

MATH 2010 – Multivariable Calculus & Matrix Algebra Professor Herron – Rensselaer Polytechnic Institute Week 1 (1/25/16 – 1/29/16) 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. I will do my best with pictures, but again, you should attend lecture so that you can draw them yourself. Function of 2 Variables - Domain: {x,y} where the function is defined - Range: set of all possible values in a function - A closed set contains its boundary, while an open set has no boundary points in its set Example 1 - ???? ????,???? = √3 − ???? − ????2 2 - Since the terms in the radical cannot be lower than 0, then: 3 − ???? − ???? ≥ 0 ???? ≥ ???? + ???? is then the domain The range is then [????,√????] Since ???? is included in the range, this is a closed set √ Example 2 - ???? ????,???? = 1 √3−???? − ????2 - Domain: ???? > ???? + ???? ???? - Since boundary points are not contained, this is an open set Application ???? - If V(x,y) is electric potential at (x,y) (???? = 2 2 2 ,???? > 0), the √???? −???? − ???? equipotentials are curves where V is constant, called level curves - Level curves are circles of varying radii Level Curves Related to Maps Level Surfaces - Imagine a function ???? ????,????,???? given temperature in space - Isothermals are surfaces (constant temperatures) ???? - ???? ????,????,???? = ???? = ???? 0 √???? + ???? + ???? =2 2 ???? → ???? + ???? + ???? = ( ) ???? √???? +???? +???? 2 ????0 ???????? ???? - Isothermals are spheres of radii????0 | Differentiation - 1 Order Partials: ???????? ???????? ???????? ???????? ???? ???? ???? = ???? ????,???? → ???????? & ???????? ;???????? & ????????; ???????????? ????,???? &) ???????????? ????,???? ) ???? = ???? ????,???? → ???? ????,???? &???? (????,????????; ???? &???? ; ???? &???????? ???? ???? All of these notations mean the same thing: first order partial derivative with respect to either x or y - In ????, y is held constant; in , x is held constant ???????? ???????? Example 3 - ???? = ???? − ???? ; take the first order partial derivative with respect to x 4 Remember to treat y as a constant; in this case, the y-term goes to 0 Take the derivative of the x-term as normal ???????? ???? = ???? ???????? ???? - Now take the partial derivative of the same function with respect to y In this case, the x-term goes to 0; derive the y-term as normal ???????? = −2???? ???????? - These partial derivatives together make a saddle surface Second Order Partials ???? ???? ???? ???? ???? ???? ???? ???? - ???? ???????????????? ????????2; ???????????? ???????? ???????????????? ;???????????? ???????? ???????????????? ; ???????? ???????? ????????2 Example 4 - ???? = ???? ???? − 3???????? + 4; take z xy First, take the partial derivative, with respect to x ????????= 3???? ???? − 3???? 2 Then, take the derivative of your answer, with respect to y ???????????? = ???????? − ???????? - Now take z oyxthe original equation ????????= ???? − 6???????? →???? ???????? = ???????? − ???????? - **If zxynd z oyxa function z = f(x,y) are continuous at a point, they are equal!** Example 5 2 2 - ???? ????,???? =) sin ???? +???? √ ???? +????2 sin???? - ???? = ????cos????, ???? = ????sin???? → ????(????cos????,????sin????) = ???? As (x,y) approaches (0,0) (r → 0), for all ????, lim = 1 (????,???? →(0,0) This means that ???? is continuous at (0,0) Tangent Planes - Review: for symmetric functions: 2 2 2 2 f(x,y) = f(y,x) √3 − ???? − ???? = √3 − ???? − ???? ???? ???? ???? Let ???? ????,???? = ???????????? ????,???? : ???? ????,???? = ???????????? ????,???? = ???????? ???? ????,???? ) - ???? = ???? ????,???? ; ???? is the slope with y held constant and ???? is the slope with x held constant ???? ???? ???? For a given point (a,b): o ???? ⃑ = ????̂ + ???? ????,???? ???? ̂ 1 ???? ̂ o ???? ⃑2= ????̂ + ????????????,???? ????) - Normal vector to the plane is: ???? = ???? ∗ ????⃑⃑ ???? ???? ????̂ ????̂ ???? ???? = | 0 1 ????????(????,????) | = ???? ????,???? ????̂ + ????????????,???? ????̂ − ???? 1 0 ????????(????,????) Example 1 2 2 2 - What is the tangent plane to ???? + ???? + ???? = 3 at (1,-1,1)? First, solve for z: 2 2 o ???? = ±√3 − ???? − ???? o Since ???? =01, take the positive square root Now take both 1 order partial derivatives and substitute in values: −???? o ???? =???? √3−???? −???? 2 ; ???? 1,−1 = −1 −???? o ???? =???? 2 2 ; ???? 1,−1 = 1 √ 3−???? −???? Now follow the above formula for the normal vector: o ???? 1????−1 ???? − 1 + ???? 1,−1 ???? + 1 − ???? − 1 = 0 ( ) o (−1 ???? − 1 − ???? + 1 − ???? − 1 = 0 ) o −???? + 1 + ???? + 1 − ???? + 1 = 0 →???? − ???? + ???? = ???? Example 2 - ???????? + ???????? + ???????? = 11, find the tangent plane at (1,2,3) 11−???????? ???? = ????+???? ???? ???? Implicit differentiation:???????? (???????? + ???????? + ???????? = ????????(11 = 0 ???????? ???????? o ???? + ???????????? + ???????????? + ???? = 0 ???? o Plug in values: 2 + 2???? ???? ???? +????3 = 0; ???? = −???? ???? ???? o ???? =???????? + ???? + ???????? + ???????? = 0 →???? ???? - − 5 (???? − 1 − 4 (???? − 2 − ???? − 3 = 0 →???????? + ???????? + ???????? = ???????? 3 3 - **If ????????(????,????) and ???? (????,????) exist and are continuous on open disk D, then ???? ????,???? is ) differentiable on D.

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