This work contains several tasks and problems of Applied Mathematics for Second Year Engineering Students.
In particular the test covers three specific fields of Mathematics: Vector calculus covers, complex analytic function and contour integrals. The solution and approach to each of the given tasks is given and laid out in consecutive succession.
Inhaltsverzeichnis (Table of Contents)
- I. Write short answer for the following Questions on Space Provided
- 1. Let r(t) = rcosti +rsintj be a circle, then find (2pts each)
- a) T(s)
- b) N(s)
- c) K(s)
- 2. Let U(x, y) = y³ — 3x²y be harmonic, then it's harmonic conjugate V(x, y) = _________?
- 3. The real and imaginary part of z = (1 + i)²i is ?
- II. Solve the following questions by showing all necessary steps
- 4. Let F(x, y, z) = (4y² + 3x²y) i + (8xy + z²) j + (11 — 2x³y) k be vector field. Then
- a) Determine F is conservative vector field or not.
- b) Find potential function of F if its conservative
- 5. Verify Green's theorem for f(x²y + x²)dx + (2xy − 4)dy where C – is the boundary of the triangle with vertices (0,0), (4,0) and (0,8) in CCW direction.
Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)
This exam paper assesses students' understanding of applied mathematics concepts, particularly in vector calculus and complex analysis. The exam covers key concepts such as finding the unit tangent, normal, and curvature vectors for a given curve, determining harmonic conjugates, exploring the real and imaginary parts of complex numbers, and applying Green's theorem.
- Vector Calculus
- Complex Analysis
- Harmonic Conjugates
- Conservative Vector Fields
- Green's Theorem
Zusammenfassung der Kapitel (Chapter Summaries)
The first section of the exam presents a series of short-answer questions, requiring students to demonstrate their understanding of basic concepts from vector calculus and complex analysis. These include finding the unit tangent, normal, and curvature vectors of a circle, finding the harmonic conjugate of a given function, and determining the real and imaginary parts of a complex number.
The second section of the exam presents more challenging problems, requiring students to apply their knowledge to specific scenarios. These include determining whether a given vector field is conservative, finding its potential function, and verifying Green's theorem for a given line integral.
Schlüsselwörter (Keywords)
This exam focuses on key concepts in vector calculus and complex analysis, including unit tangent, normal, and curvature vectors, harmonic conjugates, conservative vector fields, potential functions, and Green's theorem.
- Quote paper
- Shumatir Monijore (Author), 2020, Solved Problems of Applied Mathematics III. Final Examination of Applied Mathematics III for Second Year Engineering Students, Munich, GRIN Verlag, https://www.grin.com/document/535847
