This course covers both the theoretical foundations and practical applications of Vector Calculus. During the first week, students will learn about scalar and vector fields. In the second week, they will differentiate fields. The third week focuses on multidimensional integration and curvilinear coordinate systems. Line and surface integrals are covered in the fourth week, while the fifth week explores the fundamental theorems of vector calculus, including the gradient theorem, the divergence theorem, and Stokes' theorem. These theorems are essential for subjects in engineering such as Electromagnetism and Fluid Mechanics.
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Vector Calculus for Engineers
Ce cours fait partie de Spécialisation Mathematics for Engineers
Instructeur : Jeffrey R. Chasnov
Enseignant de premier plan
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Ce que vous apprendrez
Vectors, the dot product and cross product
The gradient, divergence, curl, and Laplacian
Multivariable integration, polar, cylindrical and spherical coordinates
Line integrals, surface integrals, the gradient theorem, the divergence theorem and Stokes' theorem
Détails à connaître
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Il y a 5 modules dans ce cours
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.
Inclus
15 vidéos28 lectures6 devoirs2 plugins
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.
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13 vidéos15 lectures5 devoirs
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.
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12 vidéos24 lectures5 devoirs
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.
Inclus
9 vidéos11 lectures4 devoirs
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.
Inclus
13 vidéos21 lectures5 devoirs
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