3d curl computation example

Khan Academy
26 May 201605:35
EducationalLearning
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TLDRThis video script offers a detailed walkthrough of calculating the curl of a vector-valued function in three-dimensional space. The function is defined with components X*Y, cos(Z), and Z^2+Y. The process involves using the del operator to take the cross product with the function, which is expanded through a determinant of a 3x3 matrix. The explanation breaks down the steps of computing partial derivatives and assembling them into the final curl result, emphasizing the importance of bookkeeping during the calculation.

Takeaways
  • 📚 The script explains how to compute the curl of a vector-valued function in three-dimensional space.
  • 🔍 It introduces a sample vector function V defined by three components: X times Y, cosine of Z, and Z squared plus Y.
  • 📐 The curl computation involves taking the cross product between the del operator and the vector function.
  • 📝 The del operator is filled with partial differential operators ∂/∂x, ∂/∂y, and ∂/∂z, which are placeholders for functions to be differentiated.
  • 🧩 The cross product is expanded using a determinant of a 3x3 matrix, which is a special case due to the inclusion of the unit vectors i, j, and k.
  • 📉 The first component of the curl is computed by taking the determinant of a sub-matrix, resulting in 1 plus sine of Z.
  • 📈 The second component, or the Y component of the curl, is zero because the partial derivatives with respect to X and Z of the given components are zero.
  • 📊 The third component of the curl, or the Z component, simplifies to negative X after taking the determinant of the relevant sub-matrix.
  • 🔢 The process requires taking six partial derivatives in total, emphasizing the importance of careful bookkeeping.
  • 📘 The script concludes by summarizing the curl of the given vector field as a function of X, Y, and Z, highlighting the components i, j, and k.
  • 🔄 The overall takeaway is that computing the curl involves understanding the cross product with the del operator and correctly applying partial derivatives.
Q & A

  • What is the definition of a vector valued function in three-dimensional space?

    -A vector valued function in three-dimensional space is a function that assigns a vector to each point in the space, typically defined by three component functions, one for each axis (X, Y, Z).

  • What is the purpose of the del operator in the context of curl computation?

    -The del operator is used to take the cross product with a vector valued function. It is filled with partial differential operators and is used to compute the curl, which is a measure of the rotation or circulation of the vector field.

  • How is the curl of a vector field computed?

    -The curl of a vector field is computed by taking the cross product between the del operator and the vector valued function, which involves expanding it into a determinant of a 3x3 matrix with unit vectors, partial differential operators, and the function components.

  • What is the significance of the unit vectors i, j, and k in the curl computation?

    -The unit vectors i, j, and k represent the directions along the X, Y, and Z axes, respectively, in three-dimensional space. They are used in the top row of the determinant matrix to compute the curl.

  • What does the second row of the determinant matrix represent in the curl computation?

    -The second row of the determinant matrix represents the partial differential operators with respect to each axis, which are placeholders for taking the derivative of the function components.

  • What is the role of the third row in the determinant matrix for curl computation?

    -The third row of the determinant matrix contains the actual components of the vector valued function, which are the functions of the variables X, Y, and Z that define the vector field.

  • How do you compute the partial derivative with respect to Y of Z squared plus Y?

    -When computing the partial derivative with respect to Y of Z squared plus Y, you treat Z as a constant. The derivative of Y with respect to Y is 1, so the partial derivative is simply 1.

  • What is the derivative of cosine Z with respect to Z?

    -The derivative of cosine Z with respect to Z is the negative of the sine of Z, which is written as -sin(Z).

  • Why is the j component of the curl result zero in the given example?

    -The j component of the curl result is zero because the partial derivatives with respect to X and Z of the function components that contribute to the j component are both zero, resulting in no contribution to the j component.

  • What is the k component of the curl in the given vector field?

    -The k component of the curl in the given vector field is -X, as it is the result of the determinant computation involving the partial derivatives with respect to X and Y of the cosine Z and X times Y components.

  • How many partial derivatives are involved in the computation of the curl for a vector field?

    -There are six partial derivatives involved in the computation of the curl for a vector field, as each component of the vector field requires two partial derivatives, one for each axis.

Outlines
00:00
📚 Introduction to Curl Computation

This paragraph introduces the concept of computing the curl of a vector valued function in three-dimensional space. The vector function V is defined with components as functions of X, Y, and Z. The explanation begins with the del operator, which is used to take the cross product with the vector function. The process involves filling the del operator with partial differential operators and then taking the cross product with the vector function. The curl is computed by expanding this into a determinant of a three-by-three matrix, where each element has a specific role. The determinant is broken down into three parts, each involving the calculation of partial derivatives of the function's components.

05:04
🔍 Detailed Steps for Curl Calculation

The second paragraph delves into the detailed steps of calculating the curl. It emphasizes the importance of correctly identifying the function's components and the del symbol's role in the computation. The process requires taking the cross product between the del operator and the function, which involves calculating six partial derivatives. The explanation outlines the method of using determinants to find the curl, highlighting the need for careful bookkeeping to ensure accuracy. The result is a vector that represents the curl of the original vector field, with each component calculated based on the specific derivatives of the function's components.

Mindmap
Keywords
💡Curl
Curl is a vector operation that describes the rotation or 'curling' of a vector field around a particular point. In the context of the video, it is used to compute the vector field's tendency to rotate around a point in three-dimensional space. The video script provides a detailed example of how to calculate the curl of a given vector function.
💡Vector Valued Function
A vector valued function is a function that maps each element from a domain to a vector in a vector space. In the video, the vector valued function V is defined in terms of X, Y, and Z, with each component of the function represented by a different mathematical expression, illustrating how the function behaves in three-dimensional space.
💡Del Operator
The del operator, represented by the symbol '∇', is a vector differential operator used in vector calculus. In the script, it is used to denote the cross product operation with the vector valued function to find the curl. The del operator is filled with partial differential operators to perform this operation.
💡Cross Product
The cross product is a binary operation on two vectors in three-dimensional space, resulting in a vector that is perpendicular to the plane containing the two vectors. In the video, the cross product is between the del operator and the vector valued function, which is expanded using a determinant to find the curl.
💡Partial Differential Operators
Partial differential operators are used to take partial derivatives of functions with respect to different variables. In the video, these operators are represented by '∂/∂X', '∂/∂Y', and '∂/∂Z', and they are used to calculate the components of the curl by taking derivatives of the vector function's components.
💡Determinant
A determinant is a scalar value that can be computed from the elements of a square matrix and has important properties in linear algebra. In the context of the video, the determinant of a 3x3 matrix is used to calculate the curl, with each element of the matrix representing a component of the cross product.
💡Unit Vectors
Unit vectors are vectors of length one, commonly used in physics and engineering to represent direction. In the video, unit vectors i, j, and k represent the directions along the X, Y, and Z axes, respectively, and are used in the determinant to form the curl.
💡Sub-determinant
A sub-determinant is a smaller determinant that is part of a larger determinant. In the video, sub-determinants are used to simplify the calculation of the curl by breaking down the 3x3 determinant into smaller, more manageable parts.
💡Partial Derivative
A partial derivative is the derivative of a function of multiple variables with respect to one of those variables, while the others are held constant. In the script, partial derivatives are taken to compute the sub-determinants for the curl calculation, such as '∂(Z^2 + Y)/∂Y'.
💡Trigonometric Functions
Trigonometric functions, such as cosine and sine, are mathematical functions of an angle. In the video, the cosine function is used in one of the components of the vector valued function, and its derivative, negative sine, appears in the curl calculation.
💡Vector Field
A vector field is a field that assigns a vector to each point in space. In the video, the vector field is defined by the vector valued function V, and the curl operation is performed to understand the field's rotational characteristics around a point.
Highlights

Introduction to computing the curl of a vector valued function in three-dimensional space.

Definition of a vector valued function V with components X*Y, cos(Z), and Z^2 + Y.

Explanation of the del operator and its role in the curl computation.

Description of the cross product between the del operator and the vector valued function.

Construction of a three-by-three matrix for curl computation using partial differential operators.

Clarification on the role of unit vectors i, j, and k in the curl determinant.

Method to compute the determinant by breaking it into three parts.

Calculation of the i component of the curl using partial derivatives with respect to Y and Z.

Derivation of the j component of the curl, emphasizing the subtraction of zero values.

Determination of the k component of the curl with partial derivatives involving X and Y.

Simplification of the curl components resulting in i: 1 + sin(Z), j: 0, and k: -X.

Emphasis on the importance of bookkeeping in the process of taking six partial derivatives.

Final expression of the curl of the vector field as a function of X, Y, and Z.

General approach to computing the curl by taking the cross product of the del symbol and the function.

Highlighting the practical applications of understanding the curl in vector calculus.

Summary of the methodical process of curl computation for educational purposes.

Transcripts

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Vector FieldsCurl Computation3D MathematicsPartial DerivativesCross ProductDel OperatorDifferential CalculusMatrix DeterminantEducational ContentMath TutorialAdvanced Math