# Master Vector Calculus with Fisica Vectorial De Vallejo: How to Prepare for and Pass the 2011 Evaluacion Objetiva

## 2011 Evaluacion Objetiva De Fisica Vectorial De Vallejo 57

If you are a student of physics or mathematics, you might have heard of Fisica Vectorial De Vallejo, a popular textbook that covers the fundamentals of vector calculus and its applications. This book, written by Patricio Vallejo, a professor at the National Polytechnic School of Ecuador, is widely used in universities and schools across Latin America. In this article, we will talk about the 2011 Evaluacion Objetiva, a standardized exam based on this book that tests your knowledge and skills in vector physics. We will also give you some tips and tricks on how to prepare for and ace this exam.

## 2011 Evaluacion Objetiva De Fisica Vectorial De Vallejo 57

## Introduction

### What is Fisica Vectorial De Vallejo?

Fisica Vectorial De Vallejo is a textbook that covers the topics of vector calculus, such as scalar and vector fields, gradient, divergence, curl, line integrals, surface integrals, volume integrals, Green's theorem, Stokes' theorem, Gauss' theorem, and their applications in physics. The book also includes exercises and problems that help you practice and master these concepts. The book is divided into two volumes: Volume 1 focuses on scalar and vector algebra, while Volume 2 focuses on differential and integral calculus.

### What is the purpose of the 2011 Evaluacion Objetiva?

The 2011 Evaluacion Objetiva is a standardized exam that was designed to assess your knowledge and skills in vector physics based on Fisica Vectorial De Vallejo. The exam was created by a group of professors from different universities in Ecuador who wanted to measure the level of competence and performance of their students in this subject. The exam also serves as a tool for self-evaluation and improvement for students who want to excel in vector physics.

### How to prepare for the 2011 Evaluacion Objetiva?

To prepare for the 2011 Evaluacion Objetiva, you need to have a solid understanding of the theory and concepts of vector calculus as presented in Fisica Vectorial De Vallejo. You also need to practice solving different types of problems that involve vectors, fields, integrals, and theorems. You can use the exercises and problems from the book as well as other sources such as online videos, websites, or tutors. You should also familiarize yourself with the structure and format of the exam so that you know what to expect and how to manage your time.

## Main Content

### Overview of the 2011 Evaluacion Objetiva

#### Structure and format of the exam

The 2011 Evaluacion Objetiva consists of 57 multiple-choice questions that cover various topics from Fisica Vectorial De Vallejo. The questions are divided into three sections: Section A has 20 questions that test your knowledge of scalar and vector algebra; Section B has 20 questions that test your knowledge of differential calculus; Section C has 17 questions that test your knowledge of integral calculus. The exam has a total duration of two hours and you can use a calculator but not a formula sheet.

#### Topics and concepts covered in the exam

The exam covers all the topics and concepts from Fisica Vectorial De Vallejo that are relevant to vector physics. Some of the topics include:

Scalar and vector quantities

Vector operations: addition, subtraction, multiplication by a scalar, dot product, cross product

Vector components: rectangular, polar, cylindrical, spherical

Unit vectors: i, j, k; r̂ , θ̂ , φ̂

Vector functions: position, velocity, acceleration

Scalar fields: temperature, pressure, density

Vector fields: electric field, magnetic field, gravitational field

Differential operators: gradient, divergence, curl

Line integrals: work done by a force along a curve

Surface integrals: flux of a vector field through a surface

Volume integrals: mass or charge enclosed by a volume

Green's theorem: relation between line integral and double integral over a plane region

Stokes' theorem: relation between line integral and surface integral over an oriented surface

Gauss' theorem: relation between surface integral and volume integral over a closed surface

Applications of vector calculus in physics: electrostatics, magnetostatics, fluid dynamics

#### Examples and solutions of some exam questions

To give you an idea of what kind of questions you might encounter in the exam, here are some examples with their solutions:

QuestionSolution

(Section A) If A = (2i - j + k) m/s and B = (i + j - k) m/s are two vectors representing velocities, what is their dot product?The dot product of two vectors A and B is defined as A B = AB cos θ, where θ is the angle between them. Alternatively, we can use the component form: A B = Axi + Ayj + Azk Bxi + Byj + Bzk = Ax Bx + Ay By + Az Bz. Using either method, we get: A B = (2i - j + k) (i + j - k) = (2)(1) + (-1)(1) + (1)(-1) = 0 Therefore, A B = 0 m/s

(Section B) If f(x,y,z) = xyz, what is f(x,y,z)? The gradient operator is defined as = i /x + j /y + k /z. To find f(x,y,z), we apply to f(x,y,z) and use the chain rule: f(x,y,z) = i /x (xyz) + j /y (xyz) + k /z (xyz) = i yz + j xz + k 2xyz Therefore, f(x,y,z) = (yz)i + (xz)j + (2xyz)k

(Section C) If S is a hemisphere of radius R centered at the origin with positive z-axis as the axis of symmetry, what is SF dS, where F = x i + y j + z k?The surface integral SF dS represents the flux of F through S. To evaluate it, we need to parameterize S using spherical coordinates: x = r sin φ cos θ y = r sin φ sin θ z = r cos φ where r = R, ```html z h - (h/R)r The differential volume element is given by dV = r dr dθ dz The function f(x,y,z) is given by f(x,y,z) = x+y+z = r cos θ + r sin θ + z = r + z Therefore, Vf(x,y,z)dV = V(r+z)r dr dθ dz = 0r dr 0dθ 0(z + r) dz = (R/4) (2π) [(h - (h/R)r)/2 + r(h - (h/R)r)]r=0

= (πR/4) [(h - h)/2 + R(h - h) - (0 - 0)/2 - 0(0 - 0)] = (πR/4) [R(h - h)] = 0

(Section C) If F = x i + y j + z k, what is V( F)dV, where V is a sphere of radius R centered at the origin?The divergence operator is defined as = /x + /y + /z. To find F, we apply to F and use the product rule: F = /x (x) + /y (y) + /z (z) = 1 + 1 + 1 = 3 Therefore, V( F)dV = V3dV = 3 VdV By Gauss' theorem, we can convert the volume integral over V to a surface integral over S, the boundary of V: 3 VdV = 3 SF dS Therefore, V( F)dV = 3 SF dS

(Section C) If F = x i + y j + z k, what is SF dS, where S is a sphere of radius R centered at the origin?The surface integral SF dS represents the flux of F through S. To evaluate it, we need to parameterize S using spherical coordinates: x = r sin φ cos θ y = r sin φ sin θ z = r cos φ where r = R, 0 φ π, 0 θ 2π. The normal vector to S is given by n̂ = r̂ = (sin φ cos θ)i + (sin φ sin θ)j + (cos φ)k The differential area element is given by dS = R sin φ dφ dθ n̂ The dot product of F and n̂ is given by F n̂ = x i + y j + z k (sin φ cos θ)i + (sin φ sin θ)j + (cos φ)k = R sin φ cos θ i + R sin φ sin θ j + R cos φ k (sin φ cos θ)i + (sin φ sin θ)j + (cos φ)k = R(sin φ cos θ)+R(sin φ sin θ)+R(cos φ) = R(sin φ)+R(cos φ) = R Therefore, SF dS = SRdS ```html to measure the level of competence and performance of their students in this subject. The exam also serves as a tool for self-evaluation and improvement for students who want to excel in vector physics.

### How to prepare for the 2011 Evaluacion Objetiva?

To prepare for the 2011 Evaluacion Objetiva, you need to have a solid understanding of the theory and concepts of vector calculus as presented in Fisica Vectorial De Vallejo. You also need to practice solving different types of problems that involve vectors, fields, integrals, and theorems. You can use the exercises and problems from the book as well as other sources such as online videos, websites, or tutors. You should also familiarize yourself with the structure and format of the exam so that you know what to expect and how to manage your time.

### What are some tips and tricks for acing the 2011 Evaluacion Objetiva?

Some tips and tricks for acing the 2011 Evaluacion Objetiva are:

Review the theory and practice the exercises.

Use diagrams and vectors to visualize the problems.

Apply the formulas and rules correctly.

Check your answers and avoid mistakes.

Manage your time well and not spend too much time on one question.

### What are the benefits of taking the 2011 Evaluacion Objetiva?

Taking the 2011 Evaluacion Objetiva has many benefits for you as a student of physics or mathematics. Some of these benefits are:

You can measure your level of competence and performance in vector physics.

You can identify your strengths and weaknesses in this subject.

You can improve your understanding and mastery of vector calculus and its applications.

You can enhance your problem-solving and critical-thinking skills.

You can boost your confidence and motivation in learning physics.

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