Green's and Stokes' theorem relationship | Multivariable Calculus | Khan Academy

Khan Academy
18 Jun 201206:54
EducationalLearning
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TLDRThis video script explores the consistency of Stokes' theorem with previously known concepts. It begins by visualizing a vector field in the xy plane and introduces a region 'R' with a boundary 'c' traversed counterclockwise. The script then applies Stokes' theorem to relate the line integral around 'c' to the double integral over 'R' of the curl of the vector field. The curl is calculated using the determinant method, revealing that for a vector field with no z-component, the curl simplifies to the partial derivatives of the field's components. The video concludes by showing that Green's theorem is a special case of Stokes' theorem when dealing with a surface in the xy plane, thus connecting the two concepts and providing a deeper understanding of vector calculus.

Takeaways
  • ๐Ÿ“š The video script discusses the application of Stokes' theorem, aiming to show its consistency with previously covered concepts.
  • ๐Ÿ“ The script introduces a region R in the xy plane with a boundary that is traversed in a counterclockwise direction, denoted as path c.
  • ๐Ÿ” A vector field f is considered, with components only in the i and j directions and no k component, meaning it lies in the xy plane.
  • ๐ŸŒ The script visualizes the vector field f as being constant in the z direction, parallel to the xy plane when z=0.
  • โˆฎ The line integral of the vector field f over the contour c is calculated, which according to Stokes' theorem should be equivalent to the surface integral of the curl of f.
  • ๐Ÿ”„ Stokes' theorem is applied to relate the line integral to the double integral of the curl of f over the region R, with the surface integral simplified to the xy plane.
  • ๐Ÿ“ The curl of f is calculated using the determinant method, resulting in a vector with only i and j components, since there is no z dependence.
  • ๐Ÿงญ The unit normal vector n for the surface in the xy plane is identified as the k vector, pointing upwards.
  • ๐Ÿ“‰ The dot product of the curl of f with the unit normal vector k simplifies the expression to the partial derivative of q with respect to x minus the partial derivative of p with respect to y.
  • ๐ŸŒฟ The special case where the surface is in the xy plane is shown to reduce Stokes' theorem to Green's theorem, illustrating the connection between the two.
  • ๐Ÿค” The script concludes by emphasizing the consistency between Green's theorem and Stokes' theorem, providing a deeper understanding of Green's theorem in the context of vector calculus.
Q & A
  • What is the main topic of the video script?

    -The main topic of the video script is Stokes' theorem and its consistency with previously learned concepts, specifically in the context of a region in the xy plane and a vector field with no z-component.

  • What does the script illustrate by drawing axes and a region in the xy plane?

    -The script illustrates the setup for applying Stokes' theorem by defining a region R in the xy plane and its boundary c, which is traversed in a counterclockwise direction.

  • What is the significance of the vector field f in the script?

    -The vector field f is significant because it is used to demonstrate the application of Stokes' theorem. It has components that are functions of x and y only, with no z-component, which simplifies the analysis.

  • What is the role of the line integral in the context of Stokes' theorem as discussed in the script?

    -The line integral in the context of Stokes' theorem represents the integral over the contour c of the vector field f dot dr, where dr is along the contour. This is one side of the equality in Stokes' theorem.

  • How does the script relate the line integral over the contour c to the double integral over the region R?

    -The script uses Stokes' theorem to equate the line integral over the contour c to the double integral over the region R of the curl of f dot n, where n is the unit normal vector to the surface.

  • What is the curl of a vector field, and how is it calculated in the script?

    -The curl of a vector field is a measure of how much the field would cause a small object placed within it to rotate. In the script, it is calculated using the determinant of a matrix involving the unit vectors i, j, k and the partial derivatives of the vector field components with respect to x, y, and z.

  • Why does the script state that the curl of f simplifies to a specific form when the vector field has no z-component?

    -The script states that the curl simplifies because the partial derivatives with respect to z of the x and y components are zero, as they do not depend on z. This results in a curl with only an i and j component, which simplifies the analysis.

  • What is the unit normal vector n in the context of the script, and why is it important?

    -In the context of the script, the unit normal vector n is the k vector, pointing straight up from the xy plane. It is important because it is used in the double integral part of Stokes' theorem to calculate the dot product with the curl of the vector field.

  • How does the script connect Stokes' theorem to Green's theorem?

    -The script connects Stokes' theorem to Green's theorem by showing that in the special case where the surface is a flat region in the xy plane, the curl of f dot n simplifies to the form used in Green's theorem, demonstrating that Green's theorem is a special case of Stokes' theorem.

  • What is the significance of the script's conclusion that Green's theorem is a special case of Stokes' theorem?

    -The significance is that it provides a deeper understanding of the relationship between the two theorems and shows how concepts from vector calculus are interconnected, enhancing the intuition behind Green's theorem and its applications.

Outlines
00:00
๐Ÿ“š Introduction to Stokes' Theorem and Its Consistency

The script begins with a review of Stokes' theorem from the previous video and aims to explore its consistency with prior knowledge. The narrator introduces a two-dimensional region R in the xy-plane with a boundary c, which is traversed counterclockwise. A vector field f is described, having only x and y components and no z component, implying the field is confined to the xy-plane. The script then delves into the line integral of f over the contour c and how, according to Stokes' theorem, this is equivalent to the double integral of the curl of f over the region R, with the normal vector n being perpendicular to the xy-plane.

05:02
๐Ÿ” Stokes' Theorem Simplification and Connection to Green's Theorem

In this paragraph, the script simplifies the curl of the vector field f, given its components do not depend on z, resulting in a curl with only partial derivatives of the x and y components. The unit normal vector n is identified as the k vector, pointing straight up from the xy-plane. The dot product of the curl of f and the unit normal vector simplifies to a form that resembles the divergence theorem but in two dimensions. This leads to the realization that Green's theorem, a fundamental principle in vector calculus for two-dimensional regions, is a special case of Stokes' theorem when the surface is confined to the xy-plane. The script concludes by emphasizing the consistency between Green's and Stokes' theorems and how this insight provides a deeper understanding of Green's theorem.

Mindmap
Keywords
๐Ÿ’กStokes' Theorem
Stokes' Theorem is a fundamental theorem in vector calculus that relates the surface integral of a vector field's curl to the line integral of the vector field around the boundary of that surface. In the video, it is used to explore the consistency with other mathematical concepts and to demonstrate its application in a specific scenario involving a region in the xy plane.
๐Ÿ’กRegion R
Region R refers to a specific area in the xy plane that is considered in the context of the problem. It is the domain over which the vector field is defined and the line integral and surface integral are calculated. The script describes Region R as the area enclosed by a boundary that is traversed in a counterclockwise direction.
๐Ÿ’กBoundary
The boundary in the script is the perimeter or edge of Region R. It is significant because it is the path (denoted as 'c') over which the line integral is taken. The direction of traversal, counterclockwise in this case, affects the calculation of the line integral.
๐Ÿ’กVector Field
A vector field is a mathematical field where each point in space is associated with a vector. In the video, the vector field 'f' has components that are functions of x and y, with no z component, meaning it is essentially two-dimensional when viewed from the perspective of the xy plane.
๐Ÿ’กLine Integral
A line integral is an integral where the integrand is taken along a curve. In the video, the line integral is taken over the contour 'c' of the vector field 'f' dot the differential displacement vector 'dr'. This concept is central to applying Stokes' Theorem.
๐Ÿ’กSurface Integral
A surface integral generalizes the concept of a line integral to two dimensions, integrating over a surface rather than a curve. In the script, the surface integral is over the region in the xy plane, and it is related to the line integral via Stokes' Theorem.
๐Ÿ’กCurl
Curl is a vector operator that measures the rotation or 'spinning' of a vector field. In the video, the curl of the vector field 'f' is calculated by taking the determinant of a matrix involving the unit vectors i, j, and k, and the partial derivatives of the vector field components with respect to x, y, and z.
๐Ÿ’กUnit Normal Vector
The unit normal vector 'n' is a vector of magnitude one that is perpendicular to a surface. In the context of the video, since the surface is in the xy plane, the unit normal vector is k, pointing in the z direction, which is used in the surface integral involving the curl of the vector field.
๐Ÿ’กGreen's Theorem
Green's Theorem is a special case of Stokes' Theorem that applies to two-dimensional regions in the plane. In the video, it is shown that when the surface is flat and lies in the xy plane, Stokes' Theorem simplifies to Green's Theorem, which relates the line integral around a boundary to the double integral over the region enclosed by the boundary.
๐Ÿ’กPartial Derivative
A partial derivative is the derivative of a function of multiple variables with respect to one of its variables, while the others are held constant. In the video, partial derivatives are used to calculate the curl of the vector field 'f', which is essential for applying Stokes' Theorem.
๐Ÿ’กConsistency
In the context of the video, consistency refers to the harmony or agreement between different mathematical theorems, specifically between Stokes' Theorem and Green's Theorem. The video demonstrates that Green's Theorem can be seen as a special case of Stokes' Theorem, thus showing their consistency.
Highlights

Exploration of Stokes' theorem consistency with previously learned concepts.

Introduction of a region R in the xy plane with a boundary traversed counterclockwise.

Vector field f described with only i and j components, no k component.

Illustration of the vector field's uniformity across the z direction.

Explanation of the line integral over contour c of vector field f.

Introduction of Stokes' theorem relating line integral to a double integral.

Clarification that the region is a surface in the xy plane for the double integral.

Calculation of the curl of f using the determinant method.

Identification of the i component of curl f as zero due to no z dependence.

Determination of the j component of curl f, also resulting in zero.

Introduction of the unit normal vector n in the context of the xy plane.

Dot product of curl f with the unit normal vector k.

Simplification of the dot product to partial derivatives of p and q.

Reduction of the problem to Green's theorem for a surface in the xy plane.

Highlighting Green's theorem as a special case of Stokes' theorem.

Connection between the intuition of Green's theorem and the curl in a region.

Reflection on the consistency and intuitive understanding of Stokes' and Green's theorems.

Transcripts
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