Stokes example part 4: Curl and final answer | Multivariable Calculus | Khan Academy

Khan Academy
20 Jun 201206:54
EducationalLearning
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TLDRIn this instructional video, the instructor guides through the process of evaluating a line integral using Stokes' Theorem. The journey begins with calculating the curl of a vector field, simplifies it, and then proceeds to set up a double integral. The integral is evaluated step by step, highlighting the simplification of the curl to a k-component and the dot product with the parameterized vector field. The final result of the integral, which equals pi, is obtained after a series of integrations over the specified bounds, showcasing the power of Stokes' Theorem in converting a complex line integral into a more manageable surface integral.

Takeaways
  • πŸ“š The lesson is focused on evaluating a complex integral using vector calculus concepts.
  • πŸ” The process begins with calculating the curl of a vector field, denoted as 'f', using the determinant method.
  • πŸ“ The del operator, represented in a different color for clarity, is used to cross with the vector field components.
  • πŸ“ˆ The vector field 'f' is given with components in terms of 'i', 'j', and 'k', with 'x', 'y', and 'z' variables respectively.
  • 🎯 The curl calculation simplifies to a single non-zero component, which is dependent on 'y' and represented in terms of 'k'.
  • 🧩 The curl result is substituted back into the integral setup, which involves a dot product with another vector.
  • πŸ”„ The dot product simplification leads to an integral that only considers the 'k' component due to the orthogonality of the other components.
  • ∫ The integral to be evaluated is then set up in terms of parameters 'r' and 'theta', with 'r' ranging from 0 to 1 and 'theta' from 0 to 2Ο€.
  • πŸ“‰ The double integral is simplified by taking the antiderivative with respect to 'theta' first, which results in terms involving 'r' and trigonometric functions.
  • πŸ“Š After evaluating the integral over the specified ranges, the result simplifies to a single definite integral involving 'r'.
  • πŸŽ‰ The final evaluation of the integral yields the result of Ο€, showcasing the successful application of Stokes' theorem to convert a line integral into a surface integral.
Q & A

  • What is the main topic discussed in the video script?

    -The main topic discussed in the video script is the evaluation of a line integral using Stokes' Theorem, specifically by calculating the curl of a vector field and then evaluating a double integral.

  • What is the curl of a vector field?

    -The curl of a vector field is a vector operator that represents the rotation of the field. It is calculated using the determinant of a matrix involving the components of the vector field and the partial derivatives with respect to spatial coordinates.

  • What is the del operator in the context of the script?

    -The del operator, represented by the symbol 'βˆ‡', is a vector differential operator that is used to take the gradient, divergence, or curl of a scalar or vector field. In the script, it is used in the calculation of the curl.

  • What is the significance of the i, j, k components in the curl calculation?

    -The i, j, k components represent the unit vectors in the x, y, and z directions, respectively. They are used to express the vector field in three-dimensional space and are crucial in the determinant used to calculate the curl.

  • How is the vector field represented in the script?

    -The vector field in the script is represented with components -y^2 in the i direction, x in the j direction, and z^2 in the k direction.

  • What simplification occurs when calculating the curl of the given vector field?

    -The simplification occurs because the partial derivatives of the constant terms with respect to the variables they are not dependent on are zero, which leads to a significant reduction in the complexity of the curl calculation.

  • What is the result of the curl calculation in the script?

    -The result of the curl calculation simplifies to 2y in the k direction, which is represented as 2yk or k(1 + 2y).

  • What is the dot product used for in the script?

    -The dot product is used to simplify the calculation of the surface integral in Stokes' Theorem by eliminating components that are orthogonal to the direction of the curl.

  • How is the double integral set up in the script?

    -The double integral is set up with respect to the parameters r and theta, with r ranging from 0 to 1 and theta ranging from 0 to 2Ο€, involving the simplified curl expression and the parameterized vector field.

  • What is the final result of the double integral calculation in the script?

    -The final result of the double integral calculation is Ο€, which is obtained after evaluating the antiderivative with respect to theta and then integrating with respect to r from 0 to 1.

  • How does the script relate the line integral to the surface integral using Stokes' Theorem?

    -The script uses Stokes' Theorem to show that the line integral can be transformed into a surface integral over a smooth surface bounded by the path of the line integral, and then evaluates this surface integral to find the same result as the line integral.

Outlines
00:00
πŸ“š Calculating the Curl of a Vector Field

In this section, the instructor focuses on calculating the curl of a vector field F. The curl is explained as the determinant of a matrix involving the unit vectors i, j, and k, and the components of the vector field. The vector field is given with components -y^2 in the i direction, x in the j direction, and z^2 in the k direction. The curl calculation simplifies to zero in the i and j components, with a k component resulting from the partial derivatives of x and -y^2 with respect to their corresponding variables. The final expression for the curl is given in terms of the parameter y, which is later substituted with r sine theta, aligning with the cylindrical coordinate system used for the subsequent integral calculations.

05:02
πŸ“ˆ Evaluating a Double Integral Using Stokes' Theorem

The second paragraph delves into the application of Stokes' Theorem to evaluate a line integral indirectly by calculating a surface integral over a smooth boundary. The curl of the vector field, previously simplified, is dotted with a differential vector representing the surface element. The resulting expression is integrated over a cylindrical region defined by parameters r and theta, with limits from 0 to 1 and 0 to 2Ο€, respectively. The integration process involves taking antiderivatives with respect to theta first, leading to an expression that simplifies after evaluating the limits of integration. The remaining integral with respect to r is then solved, yielding the final result of pi. The instructor emphasizes the significance of this result, which was obtained through a series of calculations over several videos, and contrasts it with the potential direct evaluation of the original line integral.

Mindmap
Keywords
πŸ’‘Curl
Curl is a vector operation that describes the rotation of a vector field. In the video, the instructor evaluates the curl of a given vector field 'f' to determine its rotational properties. The curl is calculated as the determinant of a matrix involving the unit vectors i, j, k, and the partial derivatives of the vector components with respect to the spatial coordinates x, y, and z. The script demonstrates the calculation of the curl, simplifying it to a form involving the variable 'y', which is crucial for applying Stokes' theorem in the context of the video.
πŸ’‘Dot Product
The dot product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number. In the script, the dot product is used to simplify the expression involving the curl of the vector field and another vector. The dot product is calculated, and since the curl has no i or j components, the calculation focuses on the k component, which simplifies the integral that needs to be evaluated.
πŸ’‘Double Integral
A double integral is a type of integral that integrates a function over a two-dimensional region. In the video, after simplifying the expression involving the curl and the dot product, the instructor is left with a double integral to evaluate. This integral is over the variables 'r' and 'theta', representing a region in polar coordinates, and is key to finding the final result of the calculation.
πŸ’‘Del Operator
The del operator, represented as 'βˆ‡', is a vector differential operator used in vector calculus. In the script, the instructor uses the del operator in the context of calculating the curl, which involves the cross product of the del operator with the vector field. The del operator is depicted with partial derivatives with respect to x, y, and z, indicating the rate of change in each direction.
πŸ’‘Stokes' Theorem
Stokes' theorem is a fundamental theorem in vector calculus that relates the surface integral of a curl to the line integral of a vector field around the boundary of the surface. In the video, the instructor uses Stokes' theorem to transform a line integral into a surface integral, which is then evaluated to find the final result of the calculation.
πŸ’‘Partial Derivative
A partial derivative is the derivative of a function of multiple variables with respect to one of its variables, while the other variables are held constant. In the script, partial derivatives are used in the calculation of the curl, where the instructor takes the partial derivatives of the components of the vector field with respect to x, y, and z.
πŸ’‘Vector Field
A vector field is a mathematical field that assigns a vector to each point in space. In the video, the vector field 'f' is given with components in the i, j, and k directions, and the instructor evaluates its properties, specifically the curl, to understand its behavior in space.
πŸ’‘Polar Coordinates
Polar coordinates are a two-dimensional coordinate system in which each point is determined by a distance from a reference point and an angle from a reference direction. In the script, the instructor uses polar coordinates to set up the double integral, with 'r' representing the radial distance and 'theta' representing the angle.
πŸ’‘Antiderivative
An antiderivative, also known as an indefinite integral, is a function that represents the reverse process of differentiation. In the video, the instructor finds the antiderivative of the integrand with respect to 'theta' to evaluate the double integral, which simplifies the process of finding the definite integral's value.
πŸ’‘Definite Integral
A definite integral is an integral that calculates the area under a curve between two points on the x-axis. In the video, after simplifying the double integral, the instructor evaluates a definite integral from 0 to 1 with respect to 'r', which gives the final numerical result of the calculation.
πŸ’‘Surface Integral
A surface integral is an integral taken over a surface rather than along a curve or a line. In the video, the surface integral is used in the context of Stokes' theorem, where the instructor evaluates the integral over a surface bounded by the path of the original line integral.
Highlights

Introduction to the final stage of the problem-solving process, emphasizing the evaluation of the curl of vector field f.

Explanation of the curl as a determinant involving the del operator and the vector field components.

Simplification of the curl calculation by identifying constant components with respect to different variables, resulting in zeros.

Derivation of the curl of f as a simplified expression involving only the k component.

Transformation of the problem into a double integral with parameters r and theta, setting up the integral bounds.

Substitution of y with r sine theta to express the curl in terms of the integral parameters.

Dot product calculation between the simplified curl and the differential vector, focusing on the k component.

Simplification of the integral expression to a form involving only r and the sine of theta.

Setting up the double integral with the correct limits for r and theta and the simplified expression.

Integration with respect to theta, finding the antiderivative and evaluating it over the given bounds.

Simplification of the theta integral results, leading to the cancellation of terms and a clear path to the final integral.

Final integration with respect to r, finding the antiderivative and evaluating from zero to one.

Result of the integral calculation, revealing the value pi as the outcome of the double integral.

Summary of the process using Stokes' theorem to transform a line integral into a surface integral.

Emphasis on the practical application of the calculated result, highlighting the significance of the value pi.

Encouragement for viewers to also attempt direct evaluation of the line integral for a deeper understanding.

Reflection on the comprehensive work done over several videos to arrive at the final result.

Transcripts

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Stokes' TheoremLine IntegralMath TutorialVector FieldsCurl CalculationDot ProductDouble IntegralSurface IntegralParametric EquationsCalculus ConceptsEducational Content