Vector Differentiation, Integration and Their Applications - Introduction to Vectors
$ 8.58
MTH 104: VectorsIntroduction to vectors, covering vector algebra, geometry, products, vector equations of geometries, vector differentiation, integration and their applications to mechanics and differential geometry.
Curated for first-year students of engineering and physical sciences at Obafemi Awolowo University, Ile-Ife, Nigeria. Students and professionals with similar learning goal will also find this learning track useful.
Introduction to vectors, covering vector algebra, geometry, products, vector equations of geometries, vector differentiation, integration and their applications to mechanics and differential geometry. Curated for first-year students of engineering and physical sciences at Obafemi Awolowo University, Ile-Ife, Nigeria. Students and professionals with similar learning goal will also find this learning track useful.
Course Chapters
1Differentiation of Vectors
Differentiation of vector-valued functions; rules of vector differentiation; derivatives of vector products; some applications of vector differentiation.
Chapter lessons
1.The derivative
Formal definitions of differentiability and the derivative of a vector-valued function.
2.Rules
Rules of differentiation for vector-valued functions.
3.Triple products
Differentiating the scalar and vector triple products of vector-valued functions.
4.Worked examples (1)
Worked examples on the derivatives of vector-valued functions.
5.Worked examples (2)
More worked examples on the derivatives of vector-valued functions.
6.Worked examples (3)
More worked examples on the derivatives of vector-valued functions.
2Integration of Vectors
Integration of vector-valued functions; definite, indefinite and line integrals of vector-valued functions; some applications of integration of vector-valued functions.
Chapter lessons
1.The anti-derivative
How to integrate a vector-valued function.
2.Line integral
Meaning of line integral and its relation to work done by a given force.
3.Position, velocity and acceleration
Linear and angular positions, velocity and acceleration of a body, and their calculus relations.
4.Worked examples (1)
Worked examples on the anti-derivatives of vector-valued functions and their mechanical applications.
5.Worked examples (2)
More worked examples on the anti-derivatives of vector-valued functions and their mechanical applications.
6.Worked examples (3)
More worked examples on the anti-derivatives of vector-valued functions and their mechanical applications.
3Mechanics I
An introduction to applications of vectors in mechanics - forces and their resultants; equilibrium under the action of concurrent forces; work done by constant and variable forces; kinetic and potential energy; conservation of energy principle; moment of a force about a point.
Chapter lessons
1.Concurrent forces
The resultant of two or more forces acting at a point; equilibrium of a particle under the action of forces.
2.Work
Re-examining the work done by constant forces and variable forces.
3.Energy
Kinetic and potential energy; power; conservation of mechanical energy.
4.Moments
Moment of a force.
5.Worked examples (1)
Worked examples involving concurrent forces, equilibrium, work, energy, power, and moments of forces.
6.Worked examples (2)
More worked examples involving concurrent forces, equilibrium, work, energy, power, and moments of forces.
7.Worked examples (3)
More worked examples involving concurrent forces, equilibrium, work, energy, power, and moments of forces.
8.Worked examples (4)
More worked examples involving concurrent forces, equilibrium, work, energy, power, and moments of forces.
4Mechanics II
An introduction to applications of vectors in mechanics - displacements, velocities and accelerations; relative velocities and accelerations; motion of a particle in tangential and normal components; motion of a particle in radial and transverse components (polar coordinates).
Chapter lessons
1.Velocity and acceleration
Relations between the position, distance, displacement, speed, velocity and acceleration of a particle.
2.Relative velocity and acceleration
Meaning and measurement of relative position, velocity and acceleration.
3.Normal and tangential components
Defining the curvilinear motion of a particle using normal and tangential vectors.
4.Radial and transverse components
Defining the curvilinear motion of a partial using radial and transverse components.
5.Worked examples (1)
Worked examples on vector analysis of rectilinear and curvilinear particle motions.
6.Worked examples (2)
More worked examples on vector analysis of rectilinear and curvilinear particle motions.
7.Worked examples (3)
More worked examples on vector analysis of rectilinear and curvilinear particle motions.
5Mechanics III
An introduction to the applications of vectors in mechanics - motion of a particle along a path of constant radius; motion of a particle in cylindrical coordinates; motion in rotating and fixed frames.
Chapter lessons
1.Constant radius
Vector analysis of the motion of a particle along a path of constant radius.
2.Cylindrical coordinates
Vector analysis of the motion of a particle using cylindrical coordinates.
3.Rotating frames
Vector analysis of the motion of a particle relative to a rotating frame of reference..
4.Worked examples (1)
Worked examples on the vector analysis of the motion of a particle involving a path of constant radius, cylindrical coordinates or rotating frames.
5.Worked examples (2)
More worked examples on the vector analysis of the motion of a particle involving a path of constant radius, cylindrical coordinates or rotating frames.
6Differential Geometry
Arc length and curvature of parametric curves; tangential, normal and binormal vectors to a parametric curve; osculating, normal and rectifying planes to a parametric curve; Frenet-Serret formulas.
Chapter lessons
1.Arc length and curvature
Arc length, tangential vector and curvature of a parametric curve.
2.Normal and binormal vectors
Normal and binormal vectors to a parametric curve.
3.Osculating, normal and rectifying planes
Osculating, normal and rectifying planes of a parametric curve.
4.Frenet-Serret formulas
The Frenet-Serret formulas and their applications.
5.Worked examples (1)
Worked examples on introductory vector differential geometry.
6.Worked examples (2)
More worked examples on introductory vector differential geometry.
7.Worked examples (3)
More worked examples on introductory vector differential geometry.