MIA 2 - Vector Concept Review.pdf
•
0 likes•7 views
Vectors have both magnitude and direction, represented by arrows. Scalars have only magnitude, represented by a single number. Operations between vectors include addition, subtraction, and cross product, while the dot product calculates the scalar projection of one vector onto another. Matrices can represent vectors and operations include transposition, addition, subtraction, and determinant calculation. Vector operations include calculating length, adding, subtracting, and using trigonometric identities involving dot products and angles between vectors.
Report
Share
Report
Share
Download to read offline
![CONCEPT REVIEW
PENULISAN VEKTOR Ԧ
𝑎 = ത
𝑎 = ො
𝑎 = 𝒂
OPERASI MATRIKS
Ԧ
𝑔 =
𝑥
𝑦
𝑧
Ԧ
𝑔𝑇
= 𝑥 𝑦 𝑧
ℎ =
1 2
3 4
5 6
ℎ𝑇 =
1 3 5
2 4 6
Misalkan Ԧ
𝑧 =
𝑎 𝑏 𝑐
𝑑 𝑒 𝑓
𝑔 ℎ 𝑖
𝐷𝑒𝑡 Ԧ
𝑧 =
𝑎 𝑏 𝑐
𝑑 𝑒 𝑓
𝑔 ℎ 𝑖
𝐷𝑒𝑡 Ԧ
𝑓 = 𝑎. 𝑒. 𝑖 + 𝑏. 𝑓. 𝑔 + 𝑐. 𝑑. ℎ − [ 𝑔. 𝑒. 𝑐 + ℎ. 𝑓. 𝑎 + (𝑖. 𝑑. 𝑏)]
Ԧ
𝑎 = (𝑚, 𝑛, 𝑝)
𝑏 = (𝑐, 𝑑, 𝑓)
Ԧ
𝑎 =
𝑚
𝑛
𝑝
𝑏 =
𝑑
𝑒
𝑓
Misalkan:
Ԧ
𝑎 = 𝑚Ԧ
𝑖 + 𝑛Ԧ
𝑗 + 𝑝𝑘
𝑏 = 𝑑Ԧ
𝑖 + 𝑒Ԧ
𝑗 + 𝑓𝑘](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Recommended
Vectores clase2
The document discusses various topics in vector algebra including:
1. Methods for adding and subtracting vectors using graphical (parallelogram and polygon methods) and analytical methods.
2. Multiplying a vector by a scalar, which changes the magnitude but not the direction of the vector.
3. Computing the dot product of two vectors, which results in a scalar value used to determine the angle between the vectors.
Vektor part 2
Slide ini menjelaskan tentang vektor pada titik formula pembagian, 3 titik segaris dan vektor- vektor tak sejajar
Vector calculus
This document discusses two theorems: Gauss divergence theorem and Stokes theorem. It provides an example problem to verify each theorem. For Gauss divergence theorem, it calculates the surface integral of a vector function over the surfaces of a rectangle parallelepiped and shows it equals the volume integral of the divergence of the function over the volume, verifying the theorem. For Stokes theorem, it similarly calculates line and surface integrals of a vector function over a rectangular region to verify the theorem holds.
Neo_Phase_1_Vectors_HN_S10_PPT.pptx
The document discusses properties of vector products. It defines the vector product of two vectors a and b as a × b and lists some of its key properties: a × b is perpendicular to both a and b; a × b · a = 0 and a × b · b = 0. It also discusses using the vector product to find a line perpendicular to two given lines and defines the vector product in terms of its Cartesian components.
Specific topics in optimisation
1) The document discusses the envelope theorem and its application to optimization problems with parameters. The envelope theorem describes how optimal values of decision variables change with parameters.
2) It provides an example of optimizing a quadratic function with respect to a single parameter b, showing how the optimal values of x and y vary with b.
3) The envelope theorem can be applied to constrained and unconstrained optimization problems. It allows calculating the rate of change of the optimal objective value with respect to parameters.
Differential Geometry for Machine Learning
References:
Differential Geometry of Curves and Surfaces, Manfredo P. Do Carmo (2016)
Differential Geometry by Claudio Arezzo
Youtube: https://youtu.be/tKnBj7B2PSg
What is a Manifold?
Youtube: https://youtu.be/CEXSSz0gZI4
Shape analysis (MIT spring 2019) by Justin Solomon
Youtube: https://youtu.be/GEljqHZb30c
Tensor Calculus
Youtube: https://youtu.be/kGXr1SF3WmA
Manifolds: A Gentle Introduction,
Hyperbolic Geometry and Poincaré Embeddings by Brian Keng
Link: http://bjlkeng.github.io/posts/manifolds/,
http://bjlkeng.github.io/posts/hyperbolic-geometry-and-poincare-embeddings/
Statistical Learning models for Manifold-Valued measurements with application to computer vision and neuroimaging by Hyunwoo J.Kim
Vector Algebra.pptx
This document discusses vector algebra concepts including:
1. Vectors can represent quantities that have both magnitude and direction, unlike scalars which only have magnitude.
2. Common vector operations include addition, subtraction, and determining the resultant or sum of multiple vectors.
3. The dot product of two vectors produces a scalar value that can indicate whether vectors are parallel or perpendicular and define physical quantities like work and electric fields.
4. The cross product of two vectors produces a new vector that is perpendicular to the original vectors and can define quantities like angular velocity and motion in electromagnetic fields.
Basic calculus (i)
The document provides an overview of basic calculus concepts including:
- Exponents and exponent rules for multiplying, dividing, and raising to powers.
- Algebraic expressions including monomials, binomials, polynomials, and equations.
- Common identities for exponents, polynomials, trigonometric functions.
- The definition of a function as a correspondence between variables where each input has a single output.
- Examples of basic functions including power, exponential, logarithmic, and trigonometric functions.
Recommended
Vectores clase2
The document discusses various topics in vector algebra including:
1. Methods for adding and subtracting vectors using graphical (parallelogram and polygon methods) and analytical methods.
2. Multiplying a vector by a scalar, which changes the magnitude but not the direction of the vector.
3. Computing the dot product of two vectors, which results in a scalar value used to determine the angle between the vectors.
Vektor part 2
Slide ini menjelaskan tentang vektor pada titik formula pembagian, 3 titik segaris dan vektor- vektor tak sejajar
Vector calculus
This document discusses two theorems: Gauss divergence theorem and Stokes theorem. It provides an example problem to verify each theorem. For Gauss divergence theorem, it calculates the surface integral of a vector function over the surfaces of a rectangle parallelepiped and shows it equals the volume integral of the divergence of the function over the volume, verifying the theorem. For Stokes theorem, it similarly calculates line and surface integrals of a vector function over a rectangular region to verify the theorem holds.
Neo_Phase_1_Vectors_HN_S10_PPT.pptx
The document discusses properties of vector products. It defines the vector product of two vectors a and b as a × b and lists some of its key properties: a × b is perpendicular to both a and b; a × b · a = 0 and a × b · b = 0. It also discusses using the vector product to find a line perpendicular to two given lines and defines the vector product in terms of its Cartesian components.
Specific topics in optimisation
1) The document discusses the envelope theorem and its application to optimization problems with parameters. The envelope theorem describes how optimal values of decision variables change with parameters.
2) It provides an example of optimizing a quadratic function with respect to a single parameter b, showing how the optimal values of x and y vary with b.
3) The envelope theorem can be applied to constrained and unconstrained optimization problems. It allows calculating the rate of change of the optimal objective value with respect to parameters.
Differential Geometry for Machine Learning
References:
Differential Geometry of Curves and Surfaces, Manfredo P. Do Carmo (2016)
Differential Geometry by Claudio Arezzo
Youtube: https://youtu.be/tKnBj7B2PSg
What is a Manifold?
Youtube: https://youtu.be/CEXSSz0gZI4
Shape analysis (MIT spring 2019) by Justin Solomon
Youtube: https://youtu.be/GEljqHZb30c
Tensor Calculus
Youtube: https://youtu.be/kGXr1SF3WmA
Manifolds: A Gentle Introduction,
Hyperbolic Geometry and Poincaré Embeddings by Brian Keng
Link: http://bjlkeng.github.io/posts/manifolds/,
http://bjlkeng.github.io/posts/hyperbolic-geometry-and-poincare-embeddings/
Statistical Learning models for Manifold-Valued measurements with application to computer vision and neuroimaging by Hyunwoo J.Kim
Vector Algebra.pptx
This document discusses vector algebra concepts including:
1. Vectors can represent quantities that have both magnitude and direction, unlike scalars which only have magnitude.
2. Common vector operations include addition, subtraction, and determining the resultant or sum of multiple vectors.
3. The dot product of two vectors produces a scalar value that can indicate whether vectors are parallel or perpendicular and define physical quantities like work and electric fields.
4. The cross product of two vectors produces a new vector that is perpendicular to the original vectors and can define quantities like angular velocity and motion in electromagnetic fields.
Basic calculus (i)
The document provides an overview of basic calculus concepts including:
- Exponents and exponent rules for multiplying, dividing, and raising to powers.
- Algebraic expressions including monomials, binomials, polynomials, and equations.
- Common identities for exponents, polynomials, trigonometric functions.
- The definition of a function as a correspondence between variables where each input has a single output.
- Examples of basic functions including power, exponential, logarithmic, and trigonometric functions.
Semana 3. integral de una función vectorial
This document discusses vector-valued functions and their integrals. It defines the indefinite and definite integrals of a vector-valued function f(t) = (f1(t), f2(t), ..., fn(t)). It also defines the arc length of a curve C described by a vector-valued function from a to b. It introduces the unit tangent vector T(t), principal normal vector N(t), and binormal vector B(t) that form the moving frame (trihedron) along the curve. It defines the osculating plane, normal plane, and rectifying plane associated with a point f(t0) on the curve.
B.tech ii unit-4 material vector differentiation
This document discusses concepts in vector differentiation including:
1. It defines scalar and vector point functions, with examples such as temperature and fluid velocity.
2. It introduces the vector differential operator del (∇) and defines the gradient of a scalar function.
3. It explains normal and directional derivatives, and gives examples of finding the gradient and directional derivative of functions.
4. It defines the divergence of a vector function and gives examples of applying vector differentiation concepts.
Elasticity, Plasticity and elastic plastic analysis
It is actually the basis of structural engineering to study elasticity and plasticity analysis. So people who are also studying in various fields of structure and need to analyze finite element analysis also need to study this basis.
Mechanics of Quadcopter
This document discusses vector mechanics concepts such as vectors, vector components, unit vectors, vector addition, scalar and vector multiplication, linear and angular velocity, and rigid body rotation. It provides definitions and examples of these concepts as they relate to mechanical engineering and the dynamics of drones. Key topics covered include defining vectors and their components, using unit vectors to express vectors, performing scalar and vector multiplication, describing linear and angular velocity, and transforming vectors under rigid body rotation.
Orbital_Simulation (2).pptx
The document outlines the assumptions and equations needed to simulate an object's orbital motion under the gravitational forces of the Earth and Moon. It assumes the Earth and Moon are stationary and there is a fixed distance between them. Only gravitational forces are acting on the object. Using vector algebra and Newton's law of gravity, the document derives an expression for the total acceleration on the object as a function of its position relative to the Earth and Moon. It then discusses using numerical methods like Euler's method and the 4th order Runge-Kutta method to iteratively calculate the object's position and velocity over time.
g_9 - L_1 Solving Quadratic Equations.pptx
Here are the steps to solve the quadratic equations given in the seatwork:
1) x2 = 64
Take the square root of both sides:
x = ±8
2) 3x2 = 108
Take the square root of both sides:
x = ±6
3) x2 - 6 = 0
Factor: (x + 3)(x - 2) = 0
x = -3, 2
4) x2 - 21 = 0
Take the square root of both sides:
x = ±√21
5) m2 + 16m = 4
Complete the square: (m + 8)2 = 16
Take the square root of both sides:
Basic calculus (ii) recap
∂z
∂x
= f′.
∂g
∂x
∂z
∂y
= f′.
∂g
∂y
The document discusses multi-variable functions and their derivatives. It defines partial derivatives as the slope of a multi-variable function with respect to one variable, holding the other variables constant. It provides examples of calculating partial derivatives using limits and applying rules like the product and chain rules. Formulas are given for finding the partial derivatives of a function z with respect to x and y at a specific point.
Btech_II_ engineering mathematics_unit4
This document discusses vector differentiation and provides examples and exercises. Some key topics covered include:
- The definition of scalar and vector point functions
- Vector differential operator Del (∇)
- Gradient of a scalar function and its properties
- Normal and directional derivatives
- Divergence and curl of vector functions
- Examples calculating gradients, directional derivatives, and verifying identities
- Exercises involving finding normals, gradients, directional derivatives, and divergence
Rational function 11
- A rational expression is a ratio of two polynomial expressions, where the denominator is not equal to zero.
- To find the domain of a rational expression, set the denominator equal to zero and solve for values of x that make the denominator equal to zero. These values are excluded from the domain.
- To find the range, find the horizontal asymptote by comparing the degrees of the numerator and denominator. The range is all real numbers except the constant value of the horizontal asymptote.
3). work & energy (finished)
1) Work is defined as the product of the force applied and the distance moved in the direction of the force. Work can cause a change in energy but the total energy in a system remains constant.
2) There are four types of work depending on whether the force and/or distance is constant or variable. Work by a conservative force depends only on the start and end points and not the path taken.
3) Kinetic energy is the energy of motion while potential energy is stored energy due to an object's position or state. Mechanical energy is the sum of an object's kinetic and potential energies.
Vehicle Routing Problem using PSO (Particle Swarm Optimization)
The document describes using particle swarm optimization to solve a capacitated vehicle routing problem. The objective is to minimize the total distance traveled by a fleet of vehicles to service customers while meeting vehicle capacity constraints. It provides details on initializing particle positions and velocities, calculating fitness based on route distance, and iteratively updating particles to find the optimal solution. The proposed approach is demonstrated on an example problem with 10 customers, 4 vehicles of capacity 200, and distances given between all locations. Pseudocode and equations are included for implementing the particle swarm optimization algorithm to find high-quality routes for the vehicle routing problem.
Computer aided design
This document describes the derivation of the structural stiffness matrix used in structural analysis. It shows that the strain is related to displacement and geometry. The strain energy of an element is then related to the forces and displacements. Equating the partial derivative of the total potential energy to zero results in a system of equations that can be written in matrix form as KQ=F, where K is the structural stiffness matrix, Q are the nodal displacements, and F are the applied nodal forces. This generalized matrix equation is used for structural analysis.
Lesson 3: Problem Set 4
1. The document discusses vector calculus concepts including the dot product, cross product, gradient, divergence, and curl of vectors. It provides examples and explanations of how to calculate these quantities.
2. The dot product yields a scalar quantity and is used to calculate the angle between two vectors. The cross product yields a vector that is perpendicular to the plane of the two input vectors.
3. The gradient is the vector of partial derivatives of a scalar function and points in the direction of maximum increase. The divergence and curl are used to describe vector fields and involve taking partial derivatives.
Schwarzchild solution derivation
A derivation of the Schwarzchild solution is presented with all relevant information. I have used this slides to teach Schwarzchild solution at my youtube channel. Here is the link
https://www.youtube.com/watch?v=ixhgvnGQZHM&t=1635s
GATE Engineering Maths : Vector Calculus
1. This document covers key concepts in vector calculus including vector basics, vector differentiation, and vector integration. It defines concepts like position vectors, gradients, divergence, curl, line integrals, and surface integrals.
2. Formulas are provided for calculating directional derivatives, divergence, curl, line integrals, surface integrals, and theorems like Green's theorem and Gauss's divergence theorem.
3. Vector operations like dot products, cross products, and triple products are defined along with their geometric interpretations and formulas for calculation.
P2-Chp12-Vectors.pptx
This document provides an overview of vectors in 3 dimensions. It discusses finding the distance between points using Pythagoras' theorem, writing vectors in i, j, k notation, finding the magnitude of a vector, and finding the angle between a vector and the coordinate axes. It also provides examples of using vectors to solve geometric problems by drawing diagrams and comparing coefficients of vectors. The exercises involve finding distances, directions of vectors, parallel vectors, and using vector operations to find missing points to form geometric shapes like parallelograms.
GraphTransformations.pptx
1. This document discusses graph transformations of functions, including translations, stretches, and reflections. It provides rules for how modifications inside and outside the function f(x) will affect the x-values and y-values of the graph.
2. Examples are given of applying transformation rules to specific points on a graph and determining the new coordinates. The document also demonstrates sketching a transformed graph using key points.
3. An exercise section provides multiple choice and short answer questions to test understanding of describing transformations and finding coordinates of transformed points.
07-Convolution.pptx signal spectra and signal processing
The document discusses convolution, which is a mathematical operation used to describe the output of a linear time-invariant system when its input is another function. Convolution combines two sequences to produce a third sequence. It has properties like commutativity and associativity. To perform a convolution, the sequences can be directly evaluated as a summation, plotted graphically and multiplied/summed, or calculated using a slide rule method by shifting and multiplying corresponding terms.
Pricing optimization poster version 2 (1)
This document presents the methodology for analyzing demand and pricing optimization for a national soft drink brand sold through a single retailer. Market data from the retailer was divided into price tiers and demand functions were estimated using least squares regression. The demand functions were then used to formulate an optimization problem to determine the wholesale and retail prices that maximize profits for the manufacturer, retailer, and competitor. The problem was solved using the method of Lagrange multipliers to find the price levels that optimize total industry profits while satisfying the profit constraints of each player.
Simple Linear Regression
Mathematical Formulation and Simple Linear Regression.
Description of the Estimation and Testing the significance of Model Parameters
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
-------------------------------------------------------------------------------
Find out more about ISO training and certification services
Training: ISO/IEC 27001 Information Security Management System - EN | PECB
ISO/IEC 42001 Artificial Intelligence Management System - EN | PECB
General Data Protection Regulation (GDPR) - Training Courses - EN | PECB
Webinars: https://pecb.com/webinars
Article: https://pecb.com/article
-------------------------------------------------------------------------------
For more information about PECB:
Website: https://pecb.com/
LinkedIn: https://www.linkedin.com/company/pecb/
Facebook: https://www.facebook.com/PECBInternational/
Slideshare: http://www.slideshare.net/PECBCERTIFICATION
More Related Content
Similar to MIA 2 - Vector Concept Review.pdf
Semana 3. integral de una función vectorial
This document discusses vector-valued functions and their integrals. It defines the indefinite and definite integrals of a vector-valued function f(t) = (f1(t), f2(t), ..., fn(t)). It also defines the arc length of a curve C described by a vector-valued function from a to b. It introduces the unit tangent vector T(t), principal normal vector N(t), and binormal vector B(t) that form the moving frame (trihedron) along the curve. It defines the osculating plane, normal plane, and rectifying plane associated with a point f(t0) on the curve.
B.tech ii unit-4 material vector differentiation
This document discusses concepts in vector differentiation including:
1. It defines scalar and vector point functions, with examples such as temperature and fluid velocity.
2. It introduces the vector differential operator del (∇) and defines the gradient of a scalar function.
3. It explains normal and directional derivatives, and gives examples of finding the gradient and directional derivative of functions.
4. It defines the divergence of a vector function and gives examples of applying vector differentiation concepts.
Elasticity, Plasticity and elastic plastic analysis
It is actually the basis of structural engineering to study elasticity and plasticity analysis. So people who are also studying in various fields of structure and need to analyze finite element analysis also need to study this basis.
Mechanics of Quadcopter
This document discusses vector mechanics concepts such as vectors, vector components, unit vectors, vector addition, scalar and vector multiplication, linear and angular velocity, and rigid body rotation. It provides definitions and examples of these concepts as they relate to mechanical engineering and the dynamics of drones. Key topics covered include defining vectors and their components, using unit vectors to express vectors, performing scalar and vector multiplication, describing linear and angular velocity, and transforming vectors under rigid body rotation.
Orbital_Simulation (2).pptx
The document outlines the assumptions and equations needed to simulate an object's orbital motion under the gravitational forces of the Earth and Moon. It assumes the Earth and Moon are stationary and there is a fixed distance between them. Only gravitational forces are acting on the object. Using vector algebra and Newton's law of gravity, the document derives an expression for the total acceleration on the object as a function of its position relative to the Earth and Moon. It then discusses using numerical methods like Euler's method and the 4th order Runge-Kutta method to iteratively calculate the object's position and velocity over time.
g_9 - L_1 Solving Quadratic Equations.pptx
Here are the steps to solve the quadratic equations given in the seatwork:
1) x2 = 64
Take the square root of both sides:
x = ±8
2) 3x2 = 108
Take the square root of both sides:
x = ±6
3) x2 - 6 = 0
Factor: (x + 3)(x - 2) = 0
x = -3, 2
4) x2 - 21 = 0
Take the square root of both sides:
x = ±√21
5) m2 + 16m = 4
Complete the square: (m + 8)2 = 16
Take the square root of both sides:
Basic calculus (ii) recap
∂z
∂x
= f′.
∂g
∂x
∂z
∂y
= f′.
∂g
∂y
The document discusses multi-variable functions and their derivatives. It defines partial derivatives as the slope of a multi-variable function with respect to one variable, holding the other variables constant. It provides examples of calculating partial derivatives using limits and applying rules like the product and chain rules. Formulas are given for finding the partial derivatives of a function z with respect to x and y at a specific point.
Btech_II_ engineering mathematics_unit4
This document discusses vector differentiation and provides examples and exercises. Some key topics covered include:
- The definition of scalar and vector point functions
- Vector differential operator Del (∇)
- Gradient of a scalar function and its properties
- Normal and directional derivatives
- Divergence and curl of vector functions
- Examples calculating gradients, directional derivatives, and verifying identities
- Exercises involving finding normals, gradients, directional derivatives, and divergence
Rational function 11
- A rational expression is a ratio of two polynomial expressions, where the denominator is not equal to zero.
- To find the domain of a rational expression, set the denominator equal to zero and solve for values of x that make the denominator equal to zero. These values are excluded from the domain.
- To find the range, find the horizontal asymptote by comparing the degrees of the numerator and denominator. The range is all real numbers except the constant value of the horizontal asymptote.
3). work & energy (finished)
1) Work is defined as the product of the force applied and the distance moved in the direction of the force. Work can cause a change in energy but the total energy in a system remains constant.
2) There are four types of work depending on whether the force and/or distance is constant or variable. Work by a conservative force depends only on the start and end points and not the path taken.
3) Kinetic energy is the energy of motion while potential energy is stored energy due to an object's position or state. Mechanical energy is the sum of an object's kinetic and potential energies.
Vehicle Routing Problem using PSO (Particle Swarm Optimization)
The document describes using particle swarm optimization to solve a capacitated vehicle routing problem. The objective is to minimize the total distance traveled by a fleet of vehicles to service customers while meeting vehicle capacity constraints. It provides details on initializing particle positions and velocities, calculating fitness based on route distance, and iteratively updating particles to find the optimal solution. The proposed approach is demonstrated on an example problem with 10 customers, 4 vehicles of capacity 200, and distances given between all locations. Pseudocode and equations are included for implementing the particle swarm optimization algorithm to find high-quality routes for the vehicle routing problem.
Computer aided design
This document describes the derivation of the structural stiffness matrix used in structural analysis. It shows that the strain is related to displacement and geometry. The strain energy of an element is then related to the forces and displacements. Equating the partial derivative of the total potential energy to zero results in a system of equations that can be written in matrix form as KQ=F, where K is the structural stiffness matrix, Q are the nodal displacements, and F are the applied nodal forces. This generalized matrix equation is used for structural analysis.
Lesson 3: Problem Set 4
1. The document discusses vector calculus concepts including the dot product, cross product, gradient, divergence, and curl of vectors. It provides examples and explanations of how to calculate these quantities.
2. The dot product yields a scalar quantity and is used to calculate the angle between two vectors. The cross product yields a vector that is perpendicular to the plane of the two input vectors.
3. The gradient is the vector of partial derivatives of a scalar function and points in the direction of maximum increase. The divergence and curl are used to describe vector fields and involve taking partial derivatives.
Schwarzchild solution derivation
A derivation of the Schwarzchild solution is presented with all relevant information. I have used this slides to teach Schwarzchild solution at my youtube channel. Here is the link
https://www.youtube.com/watch?v=ixhgvnGQZHM&t=1635s
GATE Engineering Maths : Vector Calculus
1. This document covers key concepts in vector calculus including vector basics, vector differentiation, and vector integration. It defines concepts like position vectors, gradients, divergence, curl, line integrals, and surface integrals.
2. Formulas are provided for calculating directional derivatives, divergence, curl, line integrals, surface integrals, and theorems like Green's theorem and Gauss's divergence theorem.
3. Vector operations like dot products, cross products, and triple products are defined along with their geometric interpretations and formulas for calculation.
P2-Chp12-Vectors.pptx
This document provides an overview of vectors in 3 dimensions. It discusses finding the distance between points using Pythagoras' theorem, writing vectors in i, j, k notation, finding the magnitude of a vector, and finding the angle between a vector and the coordinate axes. It also provides examples of using vectors to solve geometric problems by drawing diagrams and comparing coefficients of vectors. The exercises involve finding distances, directions of vectors, parallel vectors, and using vector operations to find missing points to form geometric shapes like parallelograms.
GraphTransformations.pptx
1. This document discusses graph transformations of functions, including translations, stretches, and reflections. It provides rules for how modifications inside and outside the function f(x) will affect the x-values and y-values of the graph.
2. Examples are given of applying transformation rules to specific points on a graph and determining the new coordinates. The document also demonstrates sketching a transformed graph using key points.
3. An exercise section provides multiple choice and short answer questions to test understanding of describing transformations and finding coordinates of transformed points.
07-Convolution.pptx signal spectra and signal processing
The document discusses convolution, which is a mathematical operation used to describe the output of a linear time-invariant system when its input is another function. Convolution combines two sequences to produce a third sequence. It has properties like commutativity and associativity. To perform a convolution, the sequences can be directly evaluated as a summation, plotted graphically and multiplied/summed, or calculated using a slide rule method by shifting and multiplying corresponding terms.
Pricing optimization poster version 2 (1)
This document presents the methodology for analyzing demand and pricing optimization for a national soft drink brand sold through a single retailer. Market data from the retailer was divided into price tiers and demand functions were estimated using least squares regression. The demand functions were then used to formulate an optimization problem to determine the wholesale and retail prices that maximize profits for the manufacturer, retailer, and competitor. The problem was solved using the method of Lagrange multipliers to find the price levels that optimize total industry profits while satisfying the profit constraints of each player.
Simple Linear Regression
Mathematical Formulation and Simple Linear Regression.
Description of the Estimation and Testing the significance of Model Parameters
Similar to MIA 2 - Vector Concept Review.pdf (20)
Elasticity, Plasticity and elastic plastic analysis
Elasticity, Plasticity and elastic plastic analysis
Vehicle Routing Problem using PSO (Particle Swarm Optimization)
Vehicle Routing Problem using PSO (Particle Swarm Optimization)
07-Convolution.pptx signal spectra and signal processing
07-Convolution.pptx signal spectra and signal processing
Recently uploaded
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
-------------------------------------------------------------------------------
Find out more about ISO training and certification services
Training: ISO/IEC 27001 Information Security Management System - EN | PECB
ISO/IEC 42001 Artificial Intelligence Management System - EN | PECB
General Data Protection Regulation (GDPR) - Training Courses - EN | PECB
Webinars: https://pecb.com/webinars
Article: https://pecb.com/article
-------------------------------------------------------------------------------
For more information about PECB:
Website: https://pecb.com/
LinkedIn: https://www.linkedin.com/company/pecb/
Facebook: https://www.facebook.com/PECBInternational/
Slideshare: http://www.slideshare.net/PECBCERTIFICATION
CACJapan - GROUP Presentation 1- Wk 4.pdf
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
Types of Herbal Cosmetics its standardization.
Physiology and chemistry of skin and pigmentation, hairs, scalp, lips and nail, Cleansing cream, Lotions, Face powders, Face packs, Lipsticks, Bath products, soaps and baby product,
Preparation and standardization of the following : Tonic, Bleaches, Dentifrices and Mouth washes & Tooth Pastes, Cosmetics for Nails.
Introduction to AI for Nonprofits with Tapp Network
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
Your Skill Boost Masterclass: Strategies for Effective Upskilling
Your Skill Boost Masterclass: Strategies for Effective UpskillingExcellence Foundation for South Sudan
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.বাংলাদেশ অর্থনৈতিক সমীক্ষা (Economic Review) ২০২৪ UJS App.pdf
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
Assessment and Planning in Educational technology.pptx
In an education system, it is understood that assessment is only for the students, but on the other hand, the Assessment of teachers is also an important aspect of the education system that ensures teachers are providing high-quality instruction to students. The assessment process can be used to provide feedback and support for professional development, to inform decisions about teacher retention or promotion, or to evaluate teacher effectiveness for accountability purposes.
How to Manage Your Lost Opportunities in Odoo 17 CRM
Odoo 17 CRM allows us to track why we lose sales opportunities with "Lost Reasons." This helps analyze our sales process and identify areas for improvement. Here's how to configure lost reasons in Odoo 17 CRM
South African Journal of Science: Writing with integrity workshop (2024)
South African Journal of Science: Writing with integrity workshop (2024)Academy of Science of South Africa
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.PCOS corelations and management through Ayurveda.
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
DRUGS AND ITS classification slide share
Any substance (other than food) that is used to prevent, diagnose, treat, or relieve symptoms of a
disease or abnormal condition
A Survey of Techniques for Maximizing LLM Performance.pptx
A Survey of Techniques for Maximizing LLM Performance
The Diamonds of 2023-2024 in the IGRA collection
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
Advanced Java[Extra Concepts, Not Difficult].docx
This is part 2 of my Java Learning Journey. This contains Hashing, ArrayList, LinkedList, Date and Time Classes, Calendar Class and more.
RPMS TEMPLATE FOR SCHOOL YEAR 2023-2024 FOR TEACHER 1 TO TEACHER 3
RPMS Template 2023-2024 by: Irene S. Rueco
Recently uploaded (20)
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...
Introduction to AI for Nonprofits with Tapp Network
Introduction to AI for Nonprofits with Tapp Network
Your Skill Boost Masterclass: Strategies for Effective Upskilling
Your Skill Boost Masterclass: Strategies for Effective Upskilling
বাংলাদেশ অর্থনৈতিক সমীক্ষা (Economic Review) ২০২৪ UJS App.pdf
বাংলাদেশ অর্থনৈতিক সমীক্ষা (Economic Review) ২০২৪ UJS App.pdf
Assessment and Planning in Educational technology.pptx
Assessment and Planning in Educational technology.pptx
How to Manage Your Lost Opportunities in Odoo 17 CRM
How to Manage Your Lost Opportunities in Odoo 17 CRM
South African Journal of Science: Writing with integrity workshop (2024)
South African Journal of Science: Writing with integrity workshop (2024)
A Survey of Techniques for Maximizing LLM Performance.pptx
A Survey of Techniques for Maximizing LLM Performance.pptx
Liberal Approach to the Study of Indian Politics.pdf
Liberal Approach to the Study of Indian Politics.pdf
RPMS TEMPLATE FOR SCHOOL YEAR 2023-2024 FOR TEACHER 1 TO TEACHER 3
RPMS TEMPLATE FOR SCHOOL YEAR 2023-2024 FOR TEACHER 1 TO TEACHER 3
MIA 2 - Vector Concept Review.pdf
- 1. CONCEPT REVIEW VECTOR vs SCALAR Berupa nilai dengan arah tertentu Berupa nilai/besaran Ԧ 𝑎 = 𝑚 Ƹ 𝑖 + 𝑛 Ƹ 𝑗 + 𝑝 𝑘 𝑎 = 𝑥, ∀𝑥 ∈ ℝ Hasil dari operasi hitung 2 vektor: Ԧ 𝑎 + 𝑏, Ԧ 𝑎 − 𝑏, Ԧ 𝑎 × 𝑏 Hasil dari perkalian skalar vektor (vector dot product) Ԧ 𝑎 ∙ 𝑏 Hasil dari menghitung Panjang vektor Ԧ 𝑎 , Ԧ 𝑎 + 𝑏 , Ԧ 𝑎 − 𝑏 , Ԧ 𝑎 × 𝑏 , Ԧ 𝑎 ∙ 𝑏 dll
- 2. CONCEPT REVIEW PENULISAN VEKTOR Ԧ 𝑎 = ത 𝑎 = ො 𝑎 = 𝒂 OPERASI MATRIKS Ԧ 𝑔 = 𝑥 𝑦 𝑧 Ԧ 𝑔𝑇 = 𝑥 𝑦 𝑧 ℎ = 1 2 3 4 5 6 ℎ𝑇 = 1 3 5 2 4 6 Misalkan Ԧ 𝑧 = 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑔 ℎ 𝑖 𝐷𝑒𝑡 Ԧ 𝑧 = 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑔 ℎ 𝑖 𝐷𝑒𝑡 Ԧ 𝑓 = 𝑎. 𝑒. 𝑖 + 𝑏. 𝑓. 𝑔 + 𝑐. 𝑑. ℎ − [ 𝑔. 𝑒. 𝑐 + ℎ. 𝑓. 𝑎 + (𝑖. 𝑑. 𝑏)] Ԧ 𝑎 = (𝑚, 𝑛, 𝑝) 𝑏 = (𝑐, 𝑑, 𝑓) Ԧ 𝑎 = 𝑚 𝑛 𝑝 𝑏 = 𝑑 𝑒 𝑓 Misalkan: Ԧ 𝑎 = 𝑚Ԧ 𝑖 + 𝑛Ԧ 𝑗 + 𝑝𝑘 𝑏 = 𝑑Ԧ 𝑖 + 𝑒Ԧ 𝑗 + 𝑓𝑘
- 3. CONCEPT REVIEW Ԧ 𝑎 + 𝑏 2 = Ԧ 𝑎 2 + 𝑏 2 + 2 Ԧ 𝑎 𝑏 cos 𝜃 Ԧ 𝑎 + 𝑏 2 = Ԧ 𝑎 2 + 𝑏 2 + 2 Ԧ 𝑎 ⋅ 𝑏 Ԧ 𝑎 − 𝑏 2 = Ԧ 𝑎 2 + 𝑏 2 − 2 Ԧ 𝑎 𝑏 cos 𝜃 Ԧ 𝑎 + 𝑏 2 = Ԧ 𝑎 2 + 𝑏 2 − 2 Ԧ 𝑎 ⋅ 𝑏
- 7. 𝒚 = 𝐬𝐢𝐧 𝒙 𝒚 = 𝐜𝐨𝐬 𝒙