Lec 11: Differentials; chain rule | MIT 18.02 Multivariable Calculus, Fall 2007

MIT OpenCourseWare

16 Jan 200950:09

EducationalLearning

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TLDRThis video script delves into the concept of differentials and their application in multivariable calculus. It explains the difference between total and partial derivatives, using the total differential to understand how variations in multiple variables affect a function's value. The script illustrates the chain rule and its extension to scenarios with multiple intermediate variables, providing a foundation for more complex derivative calculations. The discussion also touches on justifying the product and quotient rules using multivariable calculus, and concludes with an example of switching between rectangular and polar coordinates, highlighting the utility of differentials in various mathematical contexts.

Takeaways

📚 The lecture introduces the concept of differentials for functions of several variables, emphasizing their importance in understanding how these functions behave and vary.
🔄 Differentials are not the same as differences (delta); differentials are a different kind of object with their own rules of manipulation and cannot be assigned a specific value.
📉 The total differential df accounts for all contributions that can cause the function f to change, unlike partial derivatives which consider only one variable at a time.
🔑 The notation df = f_x dx + f_y dy + f_z dz is used to represent the total differential, where f_x, f_y, and f_z are partial derivatives with respect to x, y, and z, respectively.
🤔 Differentials can be thought of as placeholders that, when replaced with actual changes in variables (delta x, delta y, delta z), provide a linear approximation for the change in the function (delta f).
⛓ The chain rule is discussed in the context of differentials, showing how to find the rate of change of a function with respect to a new variable when the original variables are themselves functions of that new variable.
📐 The script provides a method to justify the product and quotient rules of differentiation using the chain rule, offering a deeper understanding of these rules in a multivariable context.
📈 An example is given to demonstrate the application of the chain rule with a function of three variables (x, y, z) where x, y, and z are functions of a single parameter t.
📝 The process of deriving the chain rule is explained, starting from the definition of the derivative and using approximation formulas, leading to the understanding that differentials encode the relationship between variables and their effect on the function's value.
📚 The script warns against mixing notations, particularly the use of 'd' with 'delta', as they represent different concepts and should not be used interchangeably.
📈 The concept of the gradient vector is introduced as a vector whose components are the partial derivatives of a function, hinting at its utility in future lectures.

Q & A

What is the main topic discussed in the script?
-The main topic discussed in the script is differentials, specifically how they are used in multivariable calculus to understand the behavior of functions of several variables and to estimate variations in arbitrary directions.
What is the difference between a partial derivative and a total differential?
-A partial derivative is a derivative with respect to one variable while holding all other variables constant, whereas a total differential includes all contributions that can cause the value of a function to change, considering variations in all variables.
Why are differentials considered different kinds of objects compared to numbers, vectors, or matrices?
-Differentials are considered different kinds of objects because they are not numbers, vectors, or matrices. They have their own rules of manipulation and are used to express the infinitesimal changes in a function in terms of the infinitesimal changes in its variables.
What is the relationship between differentials and the chain rule?
-The relationship between differentials and the chain rule is that the chain rule can be used to find the rate of change of a function with respect to a new variable when the original variables are themselves functions of that new variable. This is done by dividing the differential of the function by the differential of the new variable.
How does the script justify the use of the chain rule with differentials?
-The script justifies the use of the chain rule with differentials by explaining that if you have a function that depends on other variables, and those variables in turn depend on a new variable, you can find the rate of change of the function with respect to the new variable by using the chain rule, which is an application of differentials.
What is the significance of the gradient vector in the context of this script?
-The gradient vector is significant because it is a vector whose components are the partial derivatives of a function. It is used to package partial derivatives together and is useful for understanding the direction of the greatest rate of increase of a function, which is a concept that will be further explored in the subsequent lecture.
What is the difference between df and delta f in the script?
-In the script, df represents the differential of the function, which is not a specific number but an expression in terms of other differentials. Delta f, on the other hand, represents the actual change in the function value, which is a numerical quantity that results from small variations in the function's variables.
How can implicit differentiation be used to find the derivative of the inverse sine function, as mentioned in the script?
-Implicit differentiation can be used by first expressing the inverse sine function as y = sin(x), and then differentiating both sides with respect to x to find dx and dy. By substituting dx with cos(y) dy, you can solve for dy/dx, which is the derivative of the inverse sine function.
What is the purpose of the example given in the script involving the function w = x^2yz with x = t, y = e^t, and z = sin(t)?
-The purpose of the example is to demonstrate the application of the chain rule to find the rate of change of the function w with respect to the parameter t, and to verify the result by directly differentiating w as a function of t using single-variable calculus.
How can the script's discussion on differentials help in understanding the product and quotient rules in single-variable calculus?
-The script discusses using differentials to justify the product and quotient rules by considering a function of two variables (u and v) that are themselves functions of a single variable t. By applying the chain rule, the script shows how the usual product and quotient rules can be derived from the perspective of multivariable calculus.

Outlines

00:00

📚 Introduction to Differentials and Implicit Differentiation

The script begins with an introduction to the concept of differentials, particularly in the context of functions with multiple variables. It emphasizes the importance of understanding how functions behave and change in various directions. The video also revisits the concept of implicit differentiation from single-variable calculus, using it as a bridge to introduce the notation and manipulation of differentials in multivariable functions. The script clarifies that differentials, denoted by 'df', are not the same as changes in the function, 'delta f', and that differentials are a unique kind of object with their own rules of manipulation.

05:02

🔍 Understanding Differentials and Their Distinction from Changes

This paragraph delves deeper into the nature of differentials, explaining that they are not numbers, vectors, or matrices, but a distinct kind of object with unique manipulation rules. The script makes a clear distinction between 'df' and 'delta f', highlighting that 'df' is not a numerical value but a representation of the infinitesimal change in the function 'f'. It also introduces the concept of differentials as placeholders in the tangent plane approximation formula, which can be used to estimate changes in 'f' when the variables 'x', 'y', and 'z' undergo small variations.

10:05

📈 The Chain Rule and Its Application to Differentials

The script introduces the chain rule in the context of differentials, showing how to find the rate of change of a function 'f' with respect to a parameter 't', when the variables 'x', 'y', and 'z' are functions of 't'. It demonstrates the chain rule through the formula 'df/dt = (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt) + (∂f/∂z)(dz/dt)', and emphasizes the difference between this exact relationship and the approximate nature of the tangent plane approximation.

15:08

⚠️ Warning Against Misuse of Differential Notation

The paragraph includes a cautionary note about the misuse of differential notation, particularly the incorrect mixing of 'd' and 'delta' notations, which are fundamentally different and should not be used interchangeably. The script warns against relying on incorrect formulas presented in some textbooks and encourages a clear understanding of the proper use of differentials.

20:21

🤔 Justifying the Chain Rule with Approximations

The script attempts to justify the validity of the chain rule using approximation formulas and the concept of limits. It explains that the chain rule can be derived by considering the changes in the function 'f' as a result of small changes in time 'delta t', and taking the limit as 'delta t' approaches zero. This approach leads to an equality that represents the derivative of 'f' with respect to 't', providing a more rigorous foundation for the chain rule.

25:23

📘 Example Application of the Chain Rule

This paragraph presents an example to illustrate the application of the chain rule. It uses a function 'w' that depends on 'x', 'y', and 'z', with each of these variables being a function of time 't'. The script shows how to calculate the derivative of 'w' with respect to 't' using the chain rule and then verifies the result by directly substituting 'x', 'y', and 'z' as functions of 't' into the original function and differentiating.

30:28

📚 Deriving Single-Variable Calculus Rules Using Multivariable Concepts

The script explores the derivation of single-variable calculus rules, such as the product and quotient rules, using multivariable calculus concepts. It demonstrates how to justify these rules by considering functions of two variables 'u' and 'v' that are themselves functions of a single variable 't', and then applying the chain rule to derive the familiar product and quotient rules.

35:30

🔗 Chain Rule with Multiple Intermediate Variables

This paragraph extends the concept of the chain rule to situations where intermediate variables are functions of additional variables. It discusses how to find the partial derivatives of a function 'w' with respect to variables 'u' and 'v' when 'x' and 'y', on which 'w' depends, are themselves functions of 'u' and 'v'. The script presents a method to express these partial derivatives without explicitly substituting the variable relationships.

40:33

📐 Understanding the Chain Rule for Polar Coordinates

The script provides an example of using the chain rule to switch between rectangular and polar coordinates. It explains how to express the derivatives of a function 'f' with respect to the polar coordinates 'r' and 'theta' by relating them to the partial derivatives with respect to 'x' and 'y'. This example illustrates the practical application of the chain rule in coordinate transformations.

45:38

📝 Final Thoughts on Differentials and Upcoming Topics

The final paragraph wraps up the discussion on differentials and previews upcoming topics. It emphasizes the importance of understanding the relationship between differentials and partial derivatives, and introduces the concept of the gradient vector, which is a vector composed of the partial derivatives of a function. The script also mentions that further exploration of these concepts will be covered in future lectures.

Mindmap

Keywords

💡Partial Derivatives

Partial derivatives are a fundamental concept in calculus that describe the rate at which a multivariable function changes with respect to one variable, while holding the other variables constant. In the script, partial derivatives are used to find minima and maxima of functions of multiple variables and to understand how functions behave with respect to changes in their variables. For example, the script mentions 'partial f over partial x dx,' which represents how the function f changes with respect to the variable x.

💡Differentials

Differentials are a mathematical construct used to represent the change in a function's value due to infinitesimally small changes in its variables. The script introduces differentials as a tool to study the variations of functions of several variables. For instance, 'df' represents the change in the function 'f' due to changes in its variables, expressed as 'df equals f sub x dx plus f sub y dy plus f sub z dz,' which is the total differential.

💡Implicit Differentiation

Implicit differentiation is a technique used when a function is defined implicitly, such as 'y = sin(x)'. It allows for the calculation of the derivative of y with respect to x without explicitly solving for y. In the script, implicit differentiation is used to derive the derivative of the inverse sine function, where 'dx equals cosine y dy' leads to 'dy over dx is one over cosine y'.

💡Total Differential

The total differential is a concept that encompasses all the contributions to the change in a function's value. It is distinguished from partial derivatives by considering all variables' changes collectively. The script explains the total differential as 'df equals f sub x dx plus f sub y dy plus f sub z dz,' highlighting its role in capturing the overall variation of the function 'f'.

💡Chain Rule

The chain rule is a fundamental principle in calculus that allows for the calculation of the derivative of a composite function. It states that the derivative of a function of another function is the product of the derivative of the outer function and the derivative of the inner function. In the script, the chain rule is applied to find the rate of change of a function with respect to a parameter 't', as in 'df over dt equals f sub x dx over dt plus f sub y dy over dt plus f sub z dz over dt'.

💡Rate of Change

The rate of change refers to the speed at which a function's value changes with respect to its variables. It is a central theme in the script, where the rate of change is discussed in various contexts, such as 'the rate of change dy by dx' and 'the rate of change of f as a function of t'. The script uses the rate of change to explain how differentials and derivatives can provide insights into the behavior of functions.

💡Infinitesimal Changes

Infinitesimal changes are changes that are so small that they approach zero. In the context of the script, infinitesimal changes are used to discuss the concept of differentials and their role in approximating the change in a function's value. The script mentions 'dx, dy, and dz' as symbols representing these changes and how they relate to the function's variation.

💡Linear Approximation

Linear approximation is a method used to estimate the value of a function at a point based on its linear behavior near that point. The script refers to linear approximation when discussing how differentials can be used to approximate the change in a function, 'delta f', as 'approximately equal to fx delta x fy delta y fz delta z'.

💡Gradient Vector

The gradient vector is a mathematical concept that represents the direction and rate of the fastest increase of a function. It is a vector whose components are the partial derivatives of the function. The script introduces the gradient vector as a tool for understanding variations in functions, using the notation '∇', and mentions its application in the problems on the p-set.

💡Polar Coordinates

Polar coordinates are a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The script uses polar coordinates as an example to illustrate the process of switching between different sets of variables and how to express derivatives in terms of these new variables, such as 'partial f over partial r' and 'partial f over partial theta'.

Highlights

Introduction to the concept of differentials for functions of several variables, emphasizing their unique rules of manipulation.

Explanation of the difference between total differentials and partial derivatives, highlighting the non-numeric nature of differentials.

Motivation for studying differentials through the lens of implicit differentiation in single variable calculus.

Derivation of the derivative of the inverse sine function using implicit differentiation as an example.

Introduction of the total differential notation and its distinction from partial derivatives.

Clarification that differentials are not the same as changes in variables (deltas), and should not be confused with them.

The use of differentials as placeholders for a tangent plane approximation in multivariable calculus.

Explanation of how to find the rate of change of a function with respect to a parameter using differentials.

The chain rule in the context of differentials, relating the rate of change of a function to its partial derivatives and the derivatives of its variables.

Warning about a mistake in the textbook regarding the mixing of 'd' and 'delta' notations.

Justification of the chain rule using differentials and the concept of limits.

Application of the chain rule to verify the product rule in single variable calculus.

Demonstration of how to derive the product and quotient rules using multivariable calculus concepts.

Example of using differentials to find the rate of change of a function with respect to multiple intermediate variables.

Explanation of the chain rule for functions dependent on multiple variables, including the use of partial derivatives.

Discussion on the limitations of simplifying partial derivatives and the importance of understanding their dependencies.

Application of differentials in switching between coordinate systems, such as rectangular and polar coordinates.

Introduction to the gradient vector as a tool for packaging partial derivatives in vector form.

Transcripts

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