Symmetry of second partial derivatives

Khan Academy

11 May 201607:02

EducationalLearning

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TLDRThis video script delves into the concept of second partial derivatives in multivariable calculus. It illustrates the process of finding first and second partial derivatives for a function like sine(x) times y squared, emphasizing the different directions of differentiation. The script highlights the surprising result that the order of taking partial derivatives does not affect the outcome, provided the second derivatives are continuous—a principle known as Schwarz's theorem. It encourages viewers to explore this property with various functions and introduces alternative notations for second partial derivatives.

Takeaways

📚 The video discusses second partial derivatives in the context of multivariable functions.
🔢 A multivariable function example is given: sine of x times y squared.
🧭 Partial derivatives can be taken with respect to each variable, treating the other as a constant.
📉 Taking the partial derivative with respect to x gives cosine x times y squared.
📈 Taking the partial derivative with respect to y gives sine x times two y.
🔄 These operations result in first partial derivatives, which are also multivariable functions.
🔢 Second partial derivatives are calculated by differentiating the first partial derivatives again.
🔄 Applying the partial derivative with respect to x to the first partial derivative with respect to x yields negative sine x times y squared.
🔄 Applying the partial derivative with respect to y to the first partial derivative with respect to x yields two y times cosine x.
🔁 The order of taking partial derivatives can affect the path but not the final result, known as the equality of mixed partial derivatives.
📝 Schwarz's theorem states that if the second partial derivatives are continuous at a point, the order of differentiation does not matter.
📘 The video suggests experimenting with different multivariable functions to understand the concept of mixed partial derivatives.
📝 Notation for second partial derivatives can be simplified using juxtaposition of the variable, e.g., ∂²f/∂x∂y or ∂f/∂x∂y.

Q & A

What is the topic of the video script?
-The video script discusses second partial derivatives in the context of multivariable functions.
What is the example multivariable function given in the script?
-The example multivariable function given is sine of x times y squared (sin(x) * y^2).
What are the two options when taking the partial derivative of a function with two variables?
-The two options are to take the partial derivative with respect to x, treating y as a constant, or with respect to y, treating x as a constant.
How is the partial derivative with respect to x of the given function calculated?
-The derivative of sin(x) is cos(x), which is then multiplied by the constant y squared, resulting in cos(x) * y^2.
How is the partial derivative with respect to y of the given function calculated?
-The derivative of y squared is 2y, and since sin(x) is treated as a constant, the result is 2y * sin(x).
What are the alternate notations for the first partial derivatives with respect to x and y?
-The alternate notations are df/dy, f_sub_y for the derivative with respect to y, and f_sub_x for the derivative with respect to x.
Why are the second partial derivatives also considered multivariable functions?
-The second partial derivatives are multivariable functions because they take in two variables and output a scalar, similar to the first partial derivatives.
What is the result of applying the partial derivative with respect to x to the partial derivative of the original function with respect to x?
-The result is -sin(x) * y^2, as the derivative of cos(x) is -sin(x) and y^2 is treated as a constant.
What is Schwarz's theorem mentioned in the script and its significance?
-Schwarz's theorem states that if the second partial derivatives of a function are continuous at a relevant point, the order in which the partial derivatives are taken does not matter, meaning ∂²f/∂x∂y = ∂²f/∂y∂x.
What is the notation used for second partial derivatives in an alternative form?
-In an alternative form, the second partial derivatives can be denoted as ∂²f/∂x² or ∂²f/∂y∂x, depending on the order of differentiation.
What is the surprising result that the script finds worth pointing out?
-The surprising result is that even though the intermediate steps are different, the final values of the second partial derivatives when computed in different orders (x then y, or y then x) are equal.
What exercise does the script suggest to better understand the concept of second partial derivatives?
-The script suggests playing around with different multivariable functions, more complicated than just a product of two separate terms, to observe the equality of second partial derivatives when computed in different orders.

Outlines

00:00

📚 Introduction to Second Partial Derivatives

This paragraph introduces the concept of second partial derivatives in multivariable calculus. The speaker begins by choosing a function, sine(x) times y squared, to demonstrate the process of taking partial derivatives with respect to two variables. The explanation covers taking the first partial derivatives with respect to x and y, highlighting the process of treating one variable as a constant while differentiating with respect to the other. The paragraph also introduces alternative notations for these derivatives. The speaker then proceeds to explain how to take second partial derivatives, either by differentiating the first partial derivative with respect to x again or by taking the derivative of the first partial derivative with respect to y. The surprising result that these second derivatives, taken in different orders, yield the same value is also discussed, emphasizing the importance of this property in multivariable calculus.

05:01

🔍 The Commutativity of Second Partial Derivatives

In this paragraph, the speaker delves into the commutativity of second partial derivatives, a significant result in multivariable calculus. It is explained that if the second partial derivatives of a function are continuous at a point, the order in which the derivatives are taken does not affect the result, as per Schwarz's theorem. This property is not universal to all functions but holds true for many practical cases. The speaker encourages the audience to experiment with different multivariable functions to observe this property in action and to understand why it occurs. Additionally, the paragraph discusses alternative notations for second partial derivatives, such as using 'partial x, x' instead of the more verbose 'partial squared f / partial x squared', which can be more convenient in certain contexts. The speaker concludes by summarizing the main points and encouraging further exploration of the topic.

Mindmap

Keywords

💡Partial Derivatives

Partial derivatives are a concept in calculus that describe the rate at which a multivariable function changes with respect to one variable, while treating the other variables as constants. In the video, the theme revolves around understanding how to calculate partial derivatives for a function like sine of x times y squared. The script provides a step-by-step explanation of how to compute the partial derivative with respect to x and y, highlighting the process as a fundamental aspect of multivariable calculus.

💡Multivariable Functions

A multivariable function is a mathematical function that depends on more than one independent variable. The script introduces a specific multivariable function, sine of x times y squared, to demonstrate the concept of partial derivatives. The function serves as the basis for the entire discussion in the video, illustrating how different variables can affect the output of a function in multiple dimensions.

💡First Partial Derivatives

The first partial derivatives are the initial step in differentiating a multivariable function with respect to one of its variables. In the script, the first partial derivatives of the function sine of x times y squared with respect to x and y are calculated. These derivatives, such as cosine x times y squared and sine x times 2y, are essential for further analysis and understanding the behavior of the function.

💡Second Partial Derivatives

Second partial derivatives are the result of differentiating a first partial derivative once more with respect to the same variable. The video explains how to compute the second partial derivatives of the given function, such as negative sine x times y squared and cosine x times 2y, which are crucial for understanding the curvature of the function in the direction of each variable.

💡Leibniz Notation

Leibniz notation is a method used in calculus to denote the derivative of a function with respect to a variable. In the video, this notation is used to express second partial derivatives, such as ∂²f/∂x², which can be initially confusing due to the order of variables in the notation. The script clarifies the meaning of this notation in the context of partial derivatives.

💡Schwarz's Theorem

Schwarz's theorem, also known as the Schwarz's inequality, is a result in calculus that states a condition under which the order of taking partial derivatives does not matter. The video mentions this theorem, explaining that if the second partial derivatives are continuous at a point, then the order in which they are taken is interchangeable. This is a key takeaway from the script, emphasizing the generality of the concept.

💡Continuous Functions

In the context of the video, continuous functions are those that have no abrupt changes in value and are defined at every point in their domain. The script refers to the continuity of second partial derivatives as a condition for the interchangeability of partial derivatives, which is a fundamental aspect of Schwarz's theorem.

💡Differentiation Order

Differentiation order refers to the sequence in which derivatives are taken with respect to different variables. The video script discusses how, under certain conditions, the order of differentiation does not affect the final result, as demonstrated by the calculation of second partial derivatives in different sequences.

💡Alternate Notation

Alternate notation is a different way of writing mathematical expressions to convey the same meaning in a more concise or clear manner. The script introduces alternate notations for second partial derivatives, such as ∂f/∂x∂x and ∂f/∂y∂x, which simplify the expression and make it easier to read and understand.

💡Scalar

A scalar is a quantity that is described by a single数值 value, as opposed to a vector, which has both magnitude and direction. In the video, the script mentions that the partial derivatives of a multivariable function output a scalar, indicating that they provide a single numerical value that represents the rate of change in one direction.

💡Exercise

In the context of the video, an exercise is a practical task or problem given to the viewer to reinforce their understanding of the concepts discussed. The script encourages the viewer to try different multivariable functions and verify the interchangeability of partial derivatives, which serves as a hands-on learning experience.

Highlights

Introduction to second partial derivatives in multivariable functions.

Explanation of a multivariable function example: sine of x times y squared.

Differentiation of the function with respect to x, treating y as a constant.

Differentiation of the function with respect to y, treating x as a constant.

Introduction of first partial derivatives and their notation.

Application of second partial derivatives to the partial derivative with respect to x.

Derivation of the second partial derivative with respect to y of the partial derivative with respect to x.

Discussion on the notation and interpretation of second partial derivatives.

Surprising result that the order of taking partial derivatives can yield the same result.

Mention of Schwarz's theorem and its conditions for the commutativity of partial derivatives.

Encouragement to explore the concept with more complex multivariable functions.

Explanation of alternate notation for second partial derivatives.

Clarification of the order of operations in alternate notation for second partial derivatives.

End of the transcript with a summary of the key points discussed.

Transcripts

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Takeaways

Q & A

What is the topic of the video script?

What is the example multivariable function given in the script?

What are the two options when taking the partial derivative of a function with two variables?

How is the partial derivative with respect to x of the given function calculated?

How is the partial derivative with respect to y of the given function calculated?

What are the alternate notations for the first partial derivatives with respect to x and y?

Why are the second partial derivatives also considered multivariable functions?

What is the result of applying the partial derivative with respect to x to the partial derivative of the original function with respect to x?

What is Schwarz's theorem mentioned in the script and its significance?

What is the notation used for second partial derivatives in an alternative form?

What is the surprising result that the script finds worth pointing out?

What exercise does the script suggest to better understand the concept of second partial derivatives?