Vector Calculus for Engineers

Go to Course: https://www.coursera.org/learn/vector-calculus-engineers

Introduction

### Course Review: Vector Calculus for Engineers on Coursera **Overview** If you're a budding engineer or simply someone interested in understanding the complexities of vector calculus, the course "Vector Calculus for Engineers" on Coursera is an excellent choice. This comprehensive course delves into both the fundamental theories and practical applications of vector calculus, providing students with crucial skills applicable in various engineering fields, physics, and computer science. **Course Structure** The course is structured into five focused modules, each building upon the prior week’s knowledge, making it accessible for learners at all levels. Here’s a detailed breakdown of what you can expect from each week: 1. **Vectors**: In the first week, students are introduced to the concept of vectors—mathematical entities characterized by both magnitude and direction. The course explains vector operations, including addition, subtraction, and multiplication through dot and cross products. You'll also explore analytical geometry related to lines and planes, setting a solid groundwork for understanding scalar and vector fields. 2. **Differentiation**: The second week dives into differentiation applied to scalar and vector fields. This segment introduces partial derivatives, the chain rule for functions of several variables, and the gradient, divergence, curl, and Laplacian. By analyzing the electromagnetic wave equation derived from Maxwell’s equations, students will see practical applications of these concepts in modern communication technologies. 3. **Integration and Curvilinear Coordinates**: Week three expands on integration to encompass functions of multiple variables, presenting double and triple integrals. Students will learn about curvilinear coordinates—polar, cylindrical, and spherical—enhancing problem-solving skills for symmetrical cases. The week culminates with understanding differential operators in variable transformations, leveraging the Jacobian. 4. **Line and Surface Integrals**: The focus shifts to integration over curves and surfaces in week four. Here, students will grasp how to calculate line integrals of fields, applying them to arc lengths and work-energy theorems. Surface integrals are introduced for scalar and vector fields, showcasing their applications in defining mass flux in fluid dynamics. 5. **Fundamental Theorems**: The final week covers the fundamental theorems of vector calculus which link differentiation and integration concepts—such as the gradient theorem, divergence theorem, and Stokes' theorem. It illustrates how these principles underpin fundamental conservation laws and the structure of physical laws, culminating in a well-rounded understanding of vector calculus applications. **Why Take This Course?** 1. **Comprehensive Curriculum**: The course material is thorough, covering both theoretical concepts and real-world applications, essential for engineering students. 2. **Clear Instruction**: The instructors break down complex topics into digestible segments, making challenging content easier to understand. 3. **Skill Applicability**: Mastery of vector calculus is crucial for various engineering disciplines—be it mechanical, civil, electrical, or aerospace. This course prepares you effectively for challenges you will face in your professional career. 4. **Flexibility**: As an online course offered on Coursera, learners can progress at their own pace, making it easy to fit into busy schedules. 5. **High Demand**: The skills gained in this course are in high demand across multiple industries, making graduates of this course highly appealing to potential employers. **Recommendation** In conclusion, if you're seeking an in-depth, application-oriented approach to vector calculus, "Vector Calculus for Engineers" on Coursera is highly recommended. With its structured format, expert instruction, and relevancy in the engineering field, this course not only equips you with fundamental knowledge but also enhances your analytical skills, preparing you for advanced engineering challenges. Embrace the world of vectors and discover the power they hold in solving complex problems in your future endeavors!

Syllabus

Vectors

Vectors are mathematical constructs that have both length and direction. We define vectors and show how to add and subtract them, and how to multiply them using the dot and cross products. We apply vectors to study the analytical geometry of lines and planes, and define the Kronecker delta and the Levi-Civita symbol to prove vector identities. Finally, we define the important concepts of scalar and vector fields.

Scalar and vector fields can be differentiated. We define the partial derivative and derive the method of least squares as a minimization problem. We learn how to use the chain rule for a function of several variables, and derive the triple product rule used in chemical engineering. We define the gradient, divergence, curl, and Laplacian. We learn some useful vector calculus identities and derive them using the Kronecker delta and Levi-Civita symbol. We use vector identities to derive the electromagnetic wave equation from Maxwell's equation in free space. Electromagnetic waves form the basis of all modern communication technologies.

Integration can be extended to functions of several variables. We learn how to perform double and triple integrals. We define curvilinear coordinates, namely polar coordinates in two dimensions, and cylindrical and spherical coordinates in three dimensions, and use them to simplify problems with circular, cylindrical or spherical symmetry. We learn how to write differential operators in curvilinear coordinates and how to change variables in multidimensional integrals using the Jacobian of the transformation.

Scalar or vector fields can be integrated over curves or surfaces. We learn how to take the line integral of a scalar field and use the line integral to compute arc lengths. We then learn how to take line integrals of vector fields by taking the dot product of the vector field with tangent unit vectors to the curve. Consideration of the line integral of a force field results in the work-energy theorem. Next, we learn how to take the surface integral of a scalar field and use the surface integral to compute surface areas. We then learn how to take the surface integral of a vector field by taking the dot product of the vector field with the normal unit vector to the surface. The surface integral of a velocity field is used to define the mass flux of a fluid through a surface.

The fundamental theorem of calculus links integration with differentiation. Here, we learn the related fundamental theorems of vector calculus. These include the gradient theorem, the divergence theorem, and Stokes' theorem. We show how these theorems are used to derive continuity equations and the law of conservation of energy. We show how to define the divergence and curl in coordinate-free form, and convert the integral version of Maxwell's equations into differential form.