Model Institute of Engineering & Technology
Course Name –Engineering Mathematics-I
Course Code – BSC-101
Lecture No – 3
Topic – Euler’s Theorem
Date –
COURSE OUTCOMES
Course Outcomes | Description | Mapping with Program Outcomes and Program Specific Outcomes |
CO1 | Apply the knowledge of calculus to plot graphs of functions and solve the problem of maxima and minima.. | 1, 2, 3, 8, 10, 12 |
CO2 | Determine the convergence/divergence of infinite series, approximation of functions using power and Taylor’s series expansion and error estimation. | 1, 2, 5, 8, 10, 12 |
CO3 | Apply the concept of definite integrals to calculate area under the curves. | 1, 2, 3, 8, 10, 12 |
CO4 | Apply matrix operations and techniques such as Gauss elimination, matrix inversion, and Cramer’s rule to solve linear systems and engineering problems. | 1, 2, 5, 8, 10, 12 |
CO5 | Demonstrate knowledge of vector space by solving associated problems | 1, 2, 5, 8, 10, 12 |
Detailed Syllabus
Section A
UNIT 1: Differential Calculus– I: Leibnitz theorem (without proof), Partial differentiation, Euler’s theorem on homogeneous functions, Asymptotes, Double points, curvature, Curve tracing in Cartesian, polar and parametric forms. (11 HRS)
UNIT 2: Differential Calculus–II: Rolle’s Theorem, Mean value Theorem, Taylor’s and Maclaurin’s series with remainder , indeterminant forms, Taylor series in two variables, Maxima and Minima of functions of two variables, method of Lagrange’s multipliers. (11 HRS)
UNIT 3: Integral Calculus: Definite integrals with important properties, differentiation under the integral sign, Gamma, Beta and error functions with simple problems, applications of definite integrals to find length, area, volume and surface area of revolutions, transformation of coordinates, double and triple integrals with simple problems. (10 HRS)
Section B
Unit 4: Matrices: Matrices, vectors: addition and scalar multiplication, matrix Multiplication, Linear systems of equations, linear Independence, rank of a matrix, determinants, Cramer’s Rule, inverse of a matrix, Gauss elimination and Gauss Jordan elimination. (15 HRS)
Unit 5: Vector Space, linear dependence of vectors, basis, dimension, Linear Transformations, range and kernel of a linear map, rank and nullity, Inverse of a linear transformation, rank nullity theorem , composition of linear maps, Eigen values, eigenvectors, symmetric, skew-symmetric, and orthogonal matrices, Eigen bases. (10 HRS)
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Assessment and Evaluation Plan
Lesson Outcomes
Students will be able to:
.
Course Outcome 1 - Delivery Plan
Course Outcomes | Topics | Blooms Taxonomy |
CO1 | Apply the knowledge of calculus to plot graphs of functions and solve the problem of maxima and minima.. | Understanding |
PARTIAL DIFFERENTIATION
What is Partial Differentiation?
What is Partial Differentiation?
What is Partial Derivative?
• A partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant.
• It is denoted as ∂f/∂x, ∂f/∂y, etc.
Examples of Partial Differentiation
1. If f(x, y) = x²y + y³, then
∂f/∂x = 2xy
∂f/∂y = x² + 3y²
2. If f(x, y) = sin(xy), then
∂f/∂x = y·cos(xy)
∂f/∂y = x·cos(xy)
HOMOGENEOUS FUNCTIONS
• A function f(x, y) is said to be homogeneous of degree n if:
f(λx, λy) = λⁿf(x, y)
• Example:
f(x, y) = x² + y² is homogeneous of degree 2
• Used to describe scale-invariant functions in various fields
EULER’S THEOREM - STATEMENT
• If z = f(x, y) is a homogeneous function of degree n, then:
x(∂z/∂x) + y(∂z/∂y) = n·z
• It relates the partial derivatives of a homogeneous function to the function itself
Reflective Questions
Summarize
Partial derivatives measure how a function changes as one variable changes, keeping others constant.
• Notation includes ∂f/∂x, ∂f/∂y, etc.
• Higher order partial derivatives involve differentiating a partial derivative again.
• They are widely used in multivariable calculus and real-world applications.
Thank You
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