Euler’s Theorem for Partial Derivatives of Homogeneous Functions | Engineering Essentials

Explore Euler’s Theorem for Partial Derivatives of Homogeneous Functions of two variables in this detailed video by Engineering Essentials. Euler's Theorem is a key result in multivariable calculus, providing a powerful tool to analyze the behavior of homogeneous functions, which have widespread applications in fields like thermodynamics, economics, and engineering.

In this video, we break down:
- Euler’s Theorem: It states that for a homogeneous function, the sum of its first-order partial derivatives, each multiplied by the corresponding variable, equals the degree of the function multiplied by the function itself.
- Homogeneous Functions: These are functions that exhibit multiplicative scaling properties, meaning that multiplying the input by a constant scales the output by a power of that constant.

This video is part of the Partial Differentiation playlist and is designed for bachelor’s degree students studying Engineering, Mathematics, and Statistics at top universities like MIT, Stanford, IITs, and NITs, as well as for students preparing for competitive exams like IIT JEE and SAT.

🔑 Keywords: Euler’s Theorem, Partial Derivatives, Homogeneous Functions, Multivariable Calculus, Engineering Mathematics
#EulersTheorem #PartialDerivatives #HomogeneousFunctions #MultivariableCalculus #EngineeringMath #IITJEE #SAT #EngineeringEssentials🔑 Keywords: Partial Derivatives, Parametric Form, Implicit Functions, Implicit Differentiation, Multivariable Calculus, Engineering Mathematics
#PartialDerivatives #ParametricFunctions #ImplicitFunctions #EngineeringMath #MultivariableCalculus #IITJEE #SAT #EngineeringEssentials
🔑 Keywords: Partial Derivatives, Limits and Continuity, Two-Variable Functions, Engineering Mathematics, Multivariable Calculus, Partial Differentiation
#PartialDerivatives #LimitsAndContinuity #MultivariableCalculus #EngineeringMath #IITJEE #SAT #EngineeringEssentials

Euler’s Theorem for Partial Derivatives of Homogeneous Functions of Two variables #maths #eranand

Partial derivatives are a fundamental concept in calculus. They are used to study the behavior of multivariable functions, which are functions of more than one variable. Partial derivatives can be used to find the slope of a tangent plane to a surface, to determine the maximum and minimum values of a function, and to solve differential equations. The most common way to denote a partial derivative is with the symbol \(\frac{\partial f}{\partial x}\), where \(f\) is the function and \(x\) is the variable with respect to which the derivative is being taken. For example, if \(f(x,y)=x^{2}+y^{2}\), then \(\frac{\partial f}{\partial x}=2x\) and \(\frac{\partial f}{\partial y}=2y\). Another way to denote a partial derivative is with the use of a subscript. For example, \(f_{x}\) and \(f_{y}\) are the partial derivatives of \(f\) with respect to \(x\) and \(y\), respectively. This notation is often used when there are multiple partial derivatives in the same equation. Partial derivatives can be used to find the slope of a tangent plane to a surface. The slope of a tangent plane is a vector that points in the direction of greatest increase of the function. To find the slope of a tangent plane to a surface defined by the function \(f(x,y,z)\), we take the partial derivatives of \(f\) with respect to \(x\), \(y\), and \(z\). The slope of the tangent plane is then given by the vector \(\left( \frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z}\right) \). Partial derivatives can also be used to determine the maximum and minimum values of a function. To find the maximum or minimum value of a function, we take the partial derivatives of the function with respect to all of the variables and set them equal to zero. We then solve this system of equations to find the critical points of the function. The critical points are the points where the function may have a maximum or minimum value. To determine whether a critical point is a maximum or minimum, we look at the second order partial derivatives of the function. If the second order partial derivatives are both positive, then the critical point is a minimum. If the second order partial derivatives are both negative, then the critical point is a maximum. If the second order partial derivatives are not both positive or negative, then the critical point is a saddle point. Partial derivatives can also be used to solve differential equations. Differential equations are equations that involve derivatives of unknown functions. To solve a differential equation, we use partial derivatives to reduce the equation to a system of ordinary differential equations. We can then solve the system of ordinary differential equations to find the unknown functions. Partial derivatives are a powerful tool that can be used to study the behavior of multivariable functions. They are used in a wide variety of fields, including physics, engineering, and economics.