Learning Objectives
Upon successful completion of this module, you will be able to:
1) Define the independent and dependent variables of the function,
2) Understand the functions of two or more variables and compute the instantaneous rate of change of such functions with respect to each variable,
3) Calculate the second-order and mixed partial derivatives,
4) Distinguish the difference between integration constants for partial derivative and regular derivative,
5) Express the determinant of order 3 in terms of three determinants of order 2 by expanding along the first row,
6) Define the cross product of two vectors using the concept of determinant,
7) Understand the Leibniz notation and prime notation of the derivatives,
8) Identify the form of separable first order differential equations,
9) Solve the given differential equation by separation of variables,
10) Find a solution of the given initial-value problem,
11) Verify the given function is a solution of the given differential equation,
12) Understand that one of the classical equations of mathematical physics is known as Laplace’s equation, a type of second-order partial differential equation,
13) Understand the use and limitations of the Bernoulli equation,
14) Apply Bernoulli’s equation for a particular streamline to solve various flow problems.
Videos
Title: Ordinary (Regular) vs Partial Derivative
Summary: This video discusses the difference between ordinary (regular) derivatives and partial derivatives, and illustrates the concept with a simple example.
Learning Objectives: After watching this video, you will be able to distinguish between ordinary and partial derivatives and compute the instantaneous rate of change of functions involving two or more variables with respect to each variable.
Transcript: Read the transcript
Title: Integration Constants
Summary: This video introduces partial integration as the reverse of partial differentiation and explains the key difference between integration constants in single-variable and multivariable functions.
Learning Objectives: After watching this video, you will be able to perform partial integration and identify how the integration constant becomes a function of the other variables when integrating multivariable functions.
Transcript: Read the transcript
Title: The Cross Product of Two Vectors
Summary: This video discusses the cross product of two vectors using a determinant of order 3, which is expressed as a combination of three determinants of order 2 by expanding along the first row.
Learning Objectives: After watching this video, you will be able to compute the cross product of two vectors using the concept of determinant.
Transcript: Read the transcript
Title: Ordinary vs Partial DE
Summary: This video defines differential equations and introduces the concepts of independent and dependent variables. It then classifies differential equations by type and discusses whether an equation is an ordinary or partial differential equation.
Learning Objectives: After watching this video, you will be able to define a differential equation and distinguish between ordinary and partial differential equations by identifying their independent and dependent variables.
Transcript: Read the transcript
Title: Notations
Summary: This video discusses various notations (Leibniz notation, prime notation, Newton’s dot notation, and subscript notation) used to write ordinary differential equations. It also defines the order of a differential equation, whether ordinary or partial.
Learning Objectives: After watching this video, you will be able to identify and use different notations for writing ordinary differential equations and determine the order of both ordinary and partial differential equations.
Transcript: Read the transcript
Title: Separable Equations
Summary: This video discusses separable ordinary differential equations, shows how to solve them by separating variables, and demonstrates the use of initial value conditions to find particular solutions.
Learning Objectives: After watching this video, you will be able to identify and solve separable differential equations and apply initial conditions to determine particular solutions.
Transcript: Read the transcript
Title: Laplace’s Equation
Summary: This video introduces Laplace’s equation and explains its role in modeling steady-state systems.
Learning Objectives: After watching this video, you will be able to recognize the form of Laplace’s equation, one of the classical partial differential equations.
Transcript: Read the transcript
Title: Derivation and Discussion of Bernoulli’s equation
Summary: This video derives and discusses Bernoulli’s equation from the conservation of energy equation, emphasizing the assumptions involved and its application along a streamline.
Learning Objectives: After watching this video, you will be able to explain the form of Bernoulli’s equation, identify its assumptions, and apply this equation along a particular streamline to solve various flow problems.
Transcript: Read the transcript
Title: Solved Example Problem – Bernoulli’s Equations
Summary: This video solves a step-by-step question illustrating how to use Bernoulli’s equation and conservation of mass.
Learning Objectives: After watching this video, you will be able to apply Bernoulli’s equation and the conservation of mass to solve flow problems, including the determination of unknown pressures and velocities.
Transcript: Read the transcript