Lecture 01: Functions of several variables

Summary

In this first lecture on multivariable calculus by IITR, the concept of functions of several variables is introduced, building on the foundational understanding of single-variable functions. The lecture explores domains, ranges, and types of mappings like one-to-one and onto functions. Extending these concepts to multiple variables, the lecture covers dependent and independent variables, and touches on domain and range determination for functions of two or more variables. Additionally, the lecture includes important definitions and examples related to open and closed regions, interior and boundary points, and the notion of contour lines and level surfaces.

Highlights

• The lecture starts with a recap of single-variable functions, setting the stage for exploring several variables 📖.
• Explains domain and range succinctly, using illustrative examples to clarify these concepts 🎯.
• The description of one-to-one and onto functions gives a clear understanding of what makes functions injective or surjective 🔍.
• The transition from single-variable to multi-variable concepts showcases how mathematical principles scale ✨.
• Emphasizes the importance of understanding dependent and independent variables when dealing with functions of several variables 🔗.
• Engages with illustrative examples to elucidate the theoretical concepts in a practical, relatable way 📊.

Key Takeaways

• Understanding the basics of functions, focusing on single-variable before moving to multivariable functions 📚.
• Clear explanations on the concepts of domain and range, essential for understanding the breadth of possible outputs given certain inputs 🌐.
• Learning about one-to-one, onto functions, and exploring mappings provides insight into function behavior 🔄.
• Introduction to multivariable functions, specifying dependent and independent variables, is crucial for further studies in this area 🎓.
• Example-driven approach helps solidify knowledge, making abstract concepts easier to grasp through practical illustrations🖇️.

Overview

The lecture opens with a nuanced introduction to the world of functions, specifically transitioning from the simpler single-variable functions to the more complex realm of functions of several variables. This sets a foundation for understanding how multivariable calculus extends beyond basic function analysis, offering greater insight into how combinations of inputs determine outputs in more sophisticated ways.

Moving deeper into the lecture, there's a detailed explanation on domains and ranges, crucial building blocks for comprehending how functions map input values to outputs. The speaker uses engaging examples to illustrate these points, ensuring that even complex ideas are accessible. Key mathematical insights such as one-to-one and onto mappings are discussed, offering clarity on how function properties affect their graph and application.

Further along, attention is given to the exploration of multiple independent variables within functions, marking a distinct shift from single-variable calculus. The session emphasizes the relationship between dependent and independent variables, providing a framework that supports subsequent lectures. Definitions of open and closed regions, as well as contour lines, enrich the lecture, giving students a comprehensive toolkit for their calculus journey.

Chapters

• 00:00 - 01:00: Introduction to Functions of Several Variables The chapter introduces the concept of functions of several variables in multivariable calculus. It begins with a basic definition and explanation of what constitutes a function of two or more variables. The introductory section sets the stage for understanding how these functions operate in mathematical contexts.
• 01:00 - 04:00: Definition of Single Variable Functions The chapter focuses on the definition of single-variable functions. It begins by discussing the concept of functions with two variables and how to determine their domain and range. The text recalls previous knowledge acquired in the eleventh or twelfth standard about single-variable functions before introducing the notion of functions of several variables. The explanation includes a mathematical notation where X is a subset of R (real numbers), suggesting that there is a function F from X to R or X to Y, where Y is also mentioned.
• 04:00 - 11:00: Domain and Range of Single Variable Functions The chapter titled 'Domain and Range of Single Variable Functions' explains the basic concept of a function in mathematics. It describes a function as a special type of relation. Specifically, a function f from set X to set Y ensures that every element X in set X is associated with an element in set Y. This definition lays the foundation for understanding how single-variable functions operate, focusing on the domain and range characteristics that dictate the inputs (domain) and possible outputs (range) within such a mathematical framework.
• 11:00 - 17:00: Introduction to Functions of Two or More Variables The transcript starts by introducing the notion of a unique relationship in multi-variable functions. It discusses a specific function Y, highlighting that for a given input X, it maps to an image Y according to some defined rule, denoted as Y = f(X). This chapter likely sets the foundation by explaining the concept of functions in higher dimensions.
• 17:00 - 22:00: Domain and Range of Two Variable Functions The chapter explains the concept of functions in terms of two variables. It begins by revisiting the definition of functions with a single variable, highlighting that such functions are specific relations where every element X in set X is associated with a unique element Y in set Y. This association is determined by a rule, for example, y is equal to f(x). The chapter uses visual representations to facilitate understanding by illustrating these functions as mappings from set X to set Y.
• 22:00 - 29:00: Domain and Range of Three Variable Functions The chapter discusses the concept of domain and range in the context of functions involving three variables. The domain consists of the set 'X', where each element 'x' has a corresponding 'y' value in set 'Y' such that y = f(x). The discussion focuses on understanding how the domain (set 'X') is defined in relation to these functions.
• 29:00 - 41:00: Interior and Boundary Points The chapter discusses the concept of interior and boundary points within the context of function domains. It emphasizes understanding the domain of a function, which is denoted as the set R where the function FX is defined. The domain specifically excludes complex numbers and conditions where the denominator equals zero, ensuring the function is valid and well-defined.
• 41:00 - 48:00: Open and Closed Regions The chapter "Open and Closed Regions" discusses the fundamental concepts related to the domain and range of a function. The domain is defined as the set of all possible input values (exterior functions) for which the function is defined. The narrative then shifts to the range, which is the set of all output values from the function. The chapter outlines how the range is constructed from the domain, providing a mathematical foundation for understanding functions.
• 48:00 - 52:00: Bounded and Unbounded Regions The chapter titled 'Bounded and Unbounded Regions' discusses the concept of functions and their ranges. It starts with the explanation of the notation Y = F(X) where X belongs to a set denoted as capital X. The collection of all resulting Y values, when Y equals F(X) for X in capital X, is termed as the 'range' of the function. This range is always a subset of another set, called capital Y. When the range is a proper subset of capital Y, meaning it doesn't cover all elements of Y, such a function is referred to as an 'into function'.
• 52:00 - 60:00: Contour Lines and Level Curves This chapter introduces the concepts of into functions and level curves. It explains that a function is called an 'into function' if its range is a proper subset of its codomain Y. Conversely, if the range of F is equal to Y, it indicates a different classification of function behavior. These foundational concepts are crucial for understanding more complex mappings and their representations.
• 60:00 - 63:00: Conclusion The chapter 'Conclusion' provides a concise overview of various types of mappings and functions, specifically focusing on one-to-one mappings. It offers a definition of one-to-one function by explaining that if you take any two distinct elements in set X, the images (or outcomes) of these elements are also distinct. The chapter emphasizes on the principle that for every distinct pair of elements in the domain capital X, the mapping must ensure their images remain distinct to qualify as one-to-one mapping.

Lecture 01: Functions of several variables Transcription

• 00:00 - 00:30 [Music] [Music] [Applause] [Music] hello friends welcome to lecture series on multivariable calculus so the first lecture deals with functions of several variables that what do you mean by function of two variable or more than
• 00:30 - 01:00 two variables how can you find domain and range of those functions now in our eleventh or twelve standard we have already deal with what do you mean by a function of single variable let us recall those definition and then we will come to functions of several variables now elect XS subset of R okay then F from X to R or X to Y suppose Y is also
• 01:00 - 01:30 subset of R then f from X to Y is called a function basically a function what a function basically means a function f from X to Y is a special type of relation in which for every X in X there
• 01:30 - 02:00 exists a unique image Y in Y by some rule say Y is equal to F X so basically
• 02:00 - 02:30 what do you mean by function of the function of a single variable it means it's a special type of relation in which you take any X any element X in X derivative ways exist a unique image Y in Y by some rule say y equal to FX so you have basically this is suppose x and say this is suppose some y this is a function from x to y
• 02:30 - 03:00 we have several elements small X in capital X some elements in X and they will always exist some Y in Y such that y equal to FX ok now this X where function is defined ok this we call as domain of the function ok this X is called domain of the function so X is all those X belongs to
• 03:00 - 03:30 R where FX is defined this we call as domain of F ok now what do you mean by range of F now now what do you know if the function is defined in simple speaking we say that function is defined if it's not a complex number and the denominator should not equal to 0 ok
• 03:30 - 04:00 that we say it a function is defined and all those exterior functions we find we call dead set as domain of the function now the image of the domain of the function on Y that set is called range of the function so what how we define the range of the function so range of f is simply all those Y belongs to Y such
• 04:00 - 04:30 that Y is equals to F X and X belongs to capital X the lecture of all those why we such that y equal to F X that X belongs to capital X is called range of the function and it is quite natural that this range of the function is always a subset of capital y ok if it is a proper subset of capital y we call such function as into function ok if
• 04:30 - 05:00 range is if range of f is proper subset of y then this is called f is called into into mapping to function okay into function and if if range of F is equal to Y okay then F is
• 05:00 - 05:30 called on to mapping or on to function we have also defined one-to-one mapping or one-to-one function or man into one function how can you find that if you take any two distinct X in X and the image concerned to that distinct X and also distinct for every pair for every distinct pair X 1 X 2 in capital X we say that the mapping is one-to-one so
• 05:30 - 06:00 how can you find one-to-one mapping 1 to 1 means F is said to be one-to-one function or mapping if for every X 1 X 2 in X X 1 not equal to X 2 implies FX 1 not equal to FX 2 that means for any two
• 06:00 - 06:30 distinct p are in capital X the corresponding image in way are also different so such mappings are called one-to-one mapping and if there exists a distinct peer in capital X such that the image cross point to those distinct that distinct peer is same then that mapping is called many to one mapping so f is so f is called many to one mapping many one
• 06:30 - 07:00 mapping or many one function if there exists X 1 X 2 in X such that X 1 not equal to X 2 implies FX 1 is equal to F X so show so such mappings are called many to one mapping so this all this we have already studied in our 10 plus standard that what our mana to one
• 07:00 - 07:30 mapping what I want to and mapping when a function is said to be in to end on to okay now suppose we have this single variable function FX is equals to a simple example say under root of say X square minus one okay now I want to find out the domain and range of this function okay so this function will not be a complex number if if X square minus one is greater than equal to zero so
• 07:30 - 08:00 this quantity should be greater than equal to zero then this function will not be a complex number so that is how we can be defined domain of a function so this implies X minus 1 and X plus 1 should be greater than equal to zero so if you take this is minus one and plus one okay this is zero when X is minus 1 and this is 0 when X is plus one you take any point in between say X equal to
• 08:00 - 08:30 zero when you put X equal to zero here it is minus one when you put X equal to zero here it is plus 1 plus 1 into minus 1 is not greater than equal to 0 so shade will not be in between minus 1 to +1 shade will be from this side and from this side that is the domain of F will be minus infinity to minus 1 Union 1 to infinity okay and what is the range
• 08:30 - 09:00 of the function now this function f will never be negative because it is an under root the square root so this will never be negative so range will always be positive that means it is from 0 to infinity so in this way we can defined range and domain of a function so there's a simple illustration of a single variable function now we come to
• 09:00 - 09:30 several variable functions now come to function of two or more than two variables now first is to variable function FX Y so let X a subset of r and y a subset of r if for every X Y belongs to X cross Y there exists a unique real numbers read according to some rule this is f X Y then F X Y is called a real valued
• 09:30 - 10:00 function of two variables x and y okay now instead of one variable we are having two variables x and y okay the same definition we extend to two variable functions that means if for every XY in X cross Y there exists a unique image Z account to sum rule says that equals to F X Y then we say that it's a function of two variables x and y now here we are involving two variables two input variables and those input
• 10:00 - 10:30 variables are called independent variable and Z which is a function of x and y which is equal to F X Y that jet is called dependent variable so basically if we have this function Z equals to F X Y so this input variables this arc also called input variables okay or independent variables
• 10:30 - 11:00 and this Z which depends on X and y because if you change the values of x and y according the value of Z will change okay and this set is called output variable or dependent variable
• 11:00 - 11:30 okay now here we are having only two variables x and y similarly we may have more than two variables say n number of variables so we can also define W is equals to F x1 x2 up to xn this is a function of n variables ok so in the next light we are having function of n
• 11:30 - 12:00 variables let t be a subset of RN of real valued function f on t is a rule that assigns our real number w equal to f of X 1 X 2 up to X n where each X I belongs to R ok now here these x i's this X 1 X 2 up to xn these are the input variables which you are also called independent variables and this W which depends on X 1 H 2 up to xn is called
• 12:00 - 12:30 output variable or dependent variable ok now how can you find domain and range of two variable functions and similarly we can extend the concept for and variable functions okay in a similar way as we did for two variable functions we can defined domain and range for two variable functions as well you see we are having a two variable functions that equal to F XY here x and y are
• 12:30 - 13:00 independent variables and Z is a dependent variable okay now domain of this function f are all those XY belongs to r2 okay all those XY belongs to R 2 such that F X Y is defined connection of all
• 13:00 - 13:30 those XY in XY plane basically where F X Y is defined we call that set as a domain of the function f now the range of the function f the range of f are all those set belongs to R such that Z equal to F XY and XY belongs to domain say
• 13:30 - 14:00 this domain is represented by D so belongs to T okay so collection of all those said collection of all those Z such that Z equal to F X Y where X Y belongs to D is called range of the function f ok so that is how we can define domain and range of two variable functions similarly we can define domain and range
• 14:00 - 14:30 for and variable functions also now let us discuss few examples based on this the first example is suppose f XY is equals to is equal to under root X square plus y square ok this is at equal to F X now what are you domain of this function now domain of this function is this quantity should be greater than
• 14:30 - 15:00 equal to 0 and since X square plus X square + y square are non-negative quantities and some will also be a non negative so this this is always non-negative for any X Y in R so we say that domain is in tan or two okay you take you take any element X Y in our to X square plus y square is always greater than equal to 0 so this
• 15:00 - 15:30 will consist of domain of the function and the range will be definitely this is under root square root quantity this will never be negative so the range will be from 0 to infinity 0 will be closed because when x and y both are 0 f it was 0 so that will be included in the range of the function ok similarly the second example suppose suppose that equals to F XY is equal to
• 15:30 - 16:00 under root of it is X minus y square so what your domain of this function domain will be X minus y whole square all those s Y all those XY in R 2 such that X minus y square is non-negative will consists of domain of this function f and range will be of course 0 to
• 16:00 - 16:30 infinity it will never be negative ok now come to 3 variable functions suppose suppose function is W is equals to F XY is it is equals to 1 upon X square plus y square plus Z square now for this problem of course denominator should not equal to 0 the domain for domain of this F X square plus y square
• 16:30 - 17:00 plus Z square should not equal to 0 so because if de no matter is 0 this function will not define so we say that the domain of the function are all those XYZ such that the denominator is not equal to 0 and this is 0 only when x y is that all are 0 so this implies X Y Z should not equal to 0 0 0 and therefore
• 17:00 - 17:30 therefore domain are all those are three are three means collection of XYZ okay excluding origin excluding origin will be the domain of F and what will be a range of F the range of F will be now
• 17:30 - 18:00 this quantity will never be zero because you because it will be zero only when this tends to infinity okay so this is never be zero so it will be zero to infinity open interval zero to infinity and it is always positive because remember it is always positive so it is from zero to infinity okay now the next example is to find domain and range this w is equal to Y is e now we know
• 18:00 - 18:30 that we know that if we have a function y equal to log X say so this log X is defined when x is greater than zero okay so this y&z there is no problem because this is a multiplication of two simple variables Y into said now this
• 18:30 - 19:00 log X is defined only when X is greater than zero so this will give half region or half a space we can say when X is greater than zero this is basically half space we should say half space because it is in all XY Z where X is greater than zero so this means half space okay and the range will be now this Y is that
• 19:00 - 19:30 X may be anything Y is Z may be anything so it is from minus or plus infinity so this will be the range of this function so these are few illustrations that how can we find domain and range of several variable functions now come to a few more definitions now first is interior point now a point X naught Y naught in the s in the XY plane is called an interior
• 19:30 - 20:00 point of s if there exists a disc centered at X naught Y naught that lies entirely in s now what does it mean that is discussed by an example now consider this region s all those XY in R 2 such that X square plus y square is less so what this region is so this
• 20:00 - 20:30 is x axis and this is y axis and this is a basically unit circle or region inside unit circle okay so this is one so this would be a circle so this is all those all those points which lies inside this circle is in s okay so asses all those
• 20:30 - 21:00 all those X Y which are inside the circle now you take any point inside this region okay so this is s basically now you take any point inside this is say you point they take this point okay now there will always exist at disk centered at this point this means circle you can say you get you can always draw disk centered at this point such that
• 21:00 - 21:30 this disk always lies totally inside this region so this point we can call as this point as interior point you see if you take this point if you take a point here say you take a point here you draw a circle or a disk center at this point they will always exist there would always exist at this Center at this point such that that this lies totally
• 21:30 - 22:00 inside this region so we again say that this point is an interior point if you take a point very close to the boundary if still there will exist at this Center at this point radius may be very small but they will always says is the disk center at this point such that that this lies totally inside this region so this point is also an interior point so that means all the
• 22:00 - 22:30 points that it have this region has all the points are interior points you take any point you take any point all the points are interior points okay now if you take a point in the boundary suppose if you take a point on the boundary now if you take if you draw a disk Center at this point then this disk no matter
• 22:30 - 23:00 whatever radius you take this this will never be entirely contained in the season because there are some points which lies outside this region so this point cannot win interior point if you take a point outside this s say here and you draw any disk center at this point may be may be a larger disk okay but the disk centre at this point will never be
• 23:00 - 23:30 contained totally inside this region so this point again is not an integer point okay so all those points which lies inside this region s are the interior points now we come to a second definition that is boundary point now point X not Y not in the region s in the XY plane is called a boundary point if every disc centered at X naught Y naught contains points that lie outside of s as
• 23:30 - 24:00 well as point at lie in s now let us discuss this by an example again by an example suppose suppose initiative this you are having now this region okay that means boundary of a unit circle so what does it mean
• 24:00 - 24:30 this boundary this is only the boundary okay now you take any point on the boundary of s say you take a point here now you take you take any disk no matter how small how large radius you are taking you take any disk centered at this point draw a disk center at this point you will always find some points outside this region and you will always
• 24:30 - 25:00 find some points on the region I mean there will I mean the intersection the intersection of this disk this disk on the compliment of s and intersection of this disk honest never be empty there will be a point there will always exist some point that lies outside this disk and like that lies on the disk so this
• 25:00 - 25:30 point is an boundary point however if you take a point in the inside region say here okay if you take a point here so there will exist a disk whose intersection with the outside edges whose intersection with inside s is empty so this this point is not a
• 25:30 - 26:00 boundary point you you take you take suppose you take this this region is okay less than equal to s so this this s contains all the point eight lies on the boundary and all the point that lies inside the region s okay now if you take a point if you take a point say here okay say here now there will exist a disc Center at this point intersection
• 26:00 - 26:30 of this this with this s is non-empty by intersection of this disc with outside s is empty so this point is not a boundary point because for every disk for every disk no matter how small the radius you are taking the intersection of this this with the region s and outside of s should not empty that a definition of
• 26:30 - 27:00 boundary point so all the points that lies on the boundary are the boundary points of s okay now based on this we can decide whether region is open or closed how can we do that how can we say this let us see a region is open if it can take consists entirely of its interior points if it consists of entirely of its interior points then we say that the region is open for example
• 27:00 - 27:30 if you see on the first figure here on the slide then this figure is an unit disk without having boundary and/or it consists of all its interior points all the points that are in that that are inside this region all all are interior points so this region we can say as an open region now region is closed if it contains all its boundary points you see
• 27:30 - 28:00 if you take a third example here this is a this contained boundary points as well as as well as interior points it contain all its boundary points you see the boundary is included and it contain all its boundary points so this region is closed so in this way we can differ we can say that whether region in r2 plane I mean in XY plane whether it is open or whether it is closed if it consists
• 28:00 - 28:30 entirely of interior points we say the region is open and if it contain all its boundary points we call its closing now there may be a region which is neither open or closed for example if you take if you focus on the first figure okay and in this figure you include only those boundary points that lies that lies on the upper half of the circle
• 28:30 - 29:00 okay that means that means you are you are you are talking on only this region this is s and on the boundary only these points are included not on the lower side of the circle okay on the boundary only these points are included now it
• 29:00 - 29:30 does not contain all its interior points because these it is these are also interior point but it is not it this set does not contain those points it is not closed and these are interior points so it is not open also because if it is open then it comes then it consent energy of course interior points so there may be some regions which are neither open or closed there may be region which are open and there may be region which are only closed now a
• 29:30 - 30:00 region may be bounded or unbounded also how can I decide that now region in the plane is bounded if it lies inside the disc of fixed radius otherwise we call it unbounded region say you have this line segment okay now you can always find a circle of fixed radius that covers this line segment I mean inside which this line segment lies you can
• 30:00 - 30:30 always find a circle of fixed radius so we can say that this line segment is a bounded region you can say you can you can take a triangle also if you take a triangle you can always find a circle of fixed radius such that this triangle lies inside the circle so we can say that this triangle is a bounded region you can take a rectangle again you can find a circle of fixed radius such that
• 30:30 - 31:00 this rectangle is totally inside this region so we can say that this rectangle is a bounded region now suppose suppose you are taking our two plane the entire r2 pin I mean the first coordinate of the XY plane suppose you are taking this region now you will you can never find the circle of fixed radius such that this entire first octave first coordinate of XY plane lies
• 31:00 - 31:30 entirely in that circle so this region is unbounded region so lines half planes etcetera are the few examples of unbounded regions now contour lines now what do you mean back on two lines so let her understand this by an example okay so what contour lines are now you take this example now here that is 75
• 31:30 - 32:00 minus X square minus y square and Z is 50 now if you draw try to draw this feeder so this is X Y and signal now when X Y both are 0 shows that it is 75 so when X Y both are 0 instead it will be 75 here
• 32:00 - 32:30 okay for any other values of x and y Z it will always be less than 75 because this is positive X square is positive Y square is positive negative signs are here so for any other values of x and y z it is always less than 75 so it is it is a parabola of something like this okay now we have
• 32:30 - 33:00 a plane Z equal to 50 we have a plane it is 75 it is 0 0 75 here suppose we have Z equal to 50 now Z equal to 50 is a plane is a is a plane in the XY plane where x and y may be anything but that is always 50 okay so it gives update it
• 33:00 - 33:30 is equal to now when Z equal to 50 when you substitute Z equal to 50 here so it is 50 is equal to 75 minus X square minus y square so this implies X square plus y square is equal to 25 so it gives a
• 33:30 - 34:00 circle basically ISO square per Y square equal 25 now add Z equal to 50 at Z equal to 50 if you draw a circle of radius 5 and Center origin if you draw a circle of radius of this type radius 5 and Center origin so this circle is basically this curve basically we can say this curve is called contour lines
• 34:00 - 34:30 this curve is basically called contour lines now this this on the XY plane this circle on the X this basically circle so this circle on the XY plane is basically called label curve of F then Z equal to
• 34:30 - 35:00 50 okay so this is how we can defined contour lines control lines is basically you have a you have a surface Z equal to F X Y plane set equal to say intersect that surface and gives a curve Dada curve basically a contour line and the and that that contour line on the that that curve on the XY plane is
• 35:00 - 35:30 basically called label curve so here also I will say the same thing I'll a plane jet equal to C interceptor surface this in the points given by a F X y equal to C this curve is called contour lines of F X y equal to C ok now here it is the label curve here it is a label curve because we have a curve here now if we have a here only two independent variables are there if initial two independent variables we are having
• 35:30 - 36:00 three independent variables so this will give a level surface okay so the collection of all points X why's that in space we are a function of three independent variables have constant value F X Y Z equal to C is called label surface of f so these are two two problems let us discuss these problems quickly first is find the equation of for a level curve of the function this jet passes through this so so what is
• 36:00 - 36:30 first example this is f XY is equal to 16 minus x square minus y square and point is 2 under root 2 under root 2 now when you substitute under 200 2 and 2 and the root 2 under root 2 to here so we will obtain it is 16 - it is 8 minus 2 that is 10 that is 6 sorry
• 36:30 - 37:00 okay that is 6 so so the label curve of this will be 16 minus X square minus y square equal to 6 and implies X square plus y square equals 2 very simple you simply substitute because it passes through this point it passes through this point means this is equal to 6 and when you substitute equal to 6 here so this will give a level curve of F at this point similarly we can find every
• 37:00 - 37:30 surface of the function this at minus 1/2 comma so that is how we can defined function of several variables domain and range of a function of two or more variables domain here gives the region and that region may be closed or may be open may not be closed or may not be open or may be bonded or may be unbounded so thank you very much [Music] [Applause] [Music]