Line Integrals (2)

This module discusses line integrals, focusing on how to integrate vector fields over curves. Topics include:

• Definition and calculation of line integrals.
• Applications in calculating work done by variable forces.
• Fluid flow (flux) over closed curves and circulation integrals.

Through numerous examples, students will understand the practical implications of line integrals in various fields.

Course Lectures

• Applications of Double Integrals

  Chris Tisdell

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  This module explores the applications of double integrals within an engineering and applied mathematics framework. It covers:

  • Calculating mass and moments of 2-dimensional thin plates.
  • Understanding the concept of the centre of mass.
  • A brief introduction to triple integrals and their applications.

  Through various examples, students will gain insight into how multiple integrals are used in real-world scenarios.

• Path Integrals - How to Integrate Over Curves

  Chris Tisdell

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  This module introduces path integrals, also known as scalar line integrals. Key points include:

  • Definition and necessity of integrating functions over curves in space.
  • Applications of path integrals in engineering and physics.
  • Examples illustrating the calculation of the centre of mass of thin springs.

  Students will learn how to approach problems involving integration over curves effectively.

• Vector Fields

  Chris Tisdell

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  This module serves as an introduction to vector fields. It covers:

  • The definition and significance of a vector field in mathematics.
  • Numerous practical examples to illustrate various vector fields.
  • The role of vector fields in understanding physical phenomena.

  Students will gain foundational knowledge essential for further study in vector calculus.

• Divergence

  Chris Tisdell

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  This module discusses the concept of divergence in vector calculus. Key aspects include:

  • The definition of divergence and its interpretation as a type of derivative.
  • Applications of divergence in physics and engineering contexts.
  • Examples to clarify the computation of divergence for different vector fields.

  Students will understand how divergence relates to the behavior of vector fields.

• Curl

  Chris Tisdell

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  This module introduces the curl of a vector field, focusing on:

  • The definition of curl and its computation methods.
  • Physical interpretation in terms of circulation density.
  • Examples that illustrate the concept of curl and its applications.

  Students will learn how curl measures the tendency of a vector field to swirl around a point.

• Line Integrals (2)

  Chris Tisdell

  Playing

  This module discusses line integrals, focusing on how to integrate vector fields over curves. Topics include:

  • Definition and calculation of line integrals.
  • Applications in calculating work done by variable forces.
  • Fluid flow (flux) over closed curves and circulation integrals.

  Through numerous examples, students will understand the practical implications of line integrals in various fields.

• Applications of Line Integrals

  Chris Tisdell

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  This module explores applications of line integrals, including:

  • Calculating work done along a path.
  • Flux in the plane over curves.
  • Circulation around curves in the plane.

  Examples will illustrate these concepts, and students will learn about the fundamental theorem of line integrals.

• Fundamental Theorem of Line Integrals

  Chris Tisdell

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  This module discusses the fundamental theorem of line integrals specifically for gradient fields. Key points include:

  • Motivation behind the theorem and its statement.
  • Proof of the theorem, highlighting its significance.
  • Examples to illustrate the practical application of the theorem.

  This foundational theorem links the concepts of gradient fields and line integrals, enhancing students' understanding of vector calculus.

• Green's Theorem

  Chris Tisdell

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  This module covers Green's theorem in the plane, discussing:

  • The relationship between double integrals and line integrals.
  • Connection between curl and circulation.
  • Gauss' divergence theorem and its relationship between divergence and flux.

  Students will see how these theorems interconnect fundamental concepts in vector calculus.

• More on Green's Theorem

  Chris Tisdell

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  This module is a continuation of Green's theorem, focusing on:

  • Interesting applications of Green's theorem.
  • Exploration of several examples to solidify understanding.
  • Discussion of proofs related to the theorem.

  Students will deepen their comprehension of Green's theorem's practical applications and theoretical foundations.

• Parametrised Surfaces

  Chris Tisdell

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  This module introduces the concept of parametrized surfaces in 3D space. Key topics include:

  • The basics of parametrizing surfaces.
  • Finding tangent and normal vectors to surfaces.
  • Examples to illustrate how surfaces can be described mathematically.

  This foundational knowledge is essential for understanding surface integrals and their applications.

• Surface Integrals

  Chris Tisdell

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  This module introduces surface integrals, a generalization of double integrals in the plane. Key features include:

  • Definition and calculation of surface integrals.
  • Applications in calculating surface area.
  • Use in determining the mass of surfaces such as cones and bowls.

  Students will learn the significance of surface integrals in various fields, including engineering and physics.

• More on Surface Integrals

  Chris Tisdell

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  This module builds upon the concept of surface integrals, further exploring their calculation methods and applications. Key points include:

  • Continuation of surface integral calculations.
  • Importance in engineering contexts, such as mass calculations.
  • More examples to enhance comprehension of surface integrals.

  Students will solidify their understanding of how surface integrals are utilized in various applications.

• Surface Integrals and Vector Fields

  Chris Tisdell

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  This module discusses surface integrals of vector fields, focusing on:

  • Integrating vector fields over surfaces in 3D space.
  • Understanding flux integrals and their applications.
  • Examples that illustrate the concepts and their significance.

  Students will learn about the critical role of surface integrals in fluid flow and electromagnetics.

• Partial Differential Equations

  Chris Tisdell

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  This module covers partial differential equations (PDEs), focusing on the heat equation. Key points include:

  • Discussion and solution of the heat equation.
  • Boundary and initial conditions relevant to the problem.
  • A step-by-step method involving separation of variables and Fourier series.

  This module provides foundational knowledge for tackling PDEs in applied mathematics.

• Separable Differential Equations

  Chris Tisdell

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  This module covers separable differential equations, focusing on:

  • The method of solving separable differential equations.
  • Real-world applications in modeling dynamic phenomena.
  • Step-by-step examples to illustrate the solution process.

  Students will gain essential skills for solving differential equations commonly encountered in various fields.