Vector field example problems Systems where one of the 2. A time-dependent vector field depends also on time. g. gravitation), Here are a set of practice problems for the Surface Integrals chapter of the Calculus III notes. \[\vec F = \left( {6{x^2} - 2x{y^2} + \frac{y}{{2\sqrt x }}} \right)\vec i - \left( {2{x^2}y - 4 - \sqrt x } \right)\vec j\] One of Vectors are useful tools for solving two-dimensional problems. If F is a vector field, then the process of dividing F by its magnitude to form unit vector Solution: magnetic flux is the scalar product of the magnetic field and the area vector with formula $\Phi_m=\vec{B}\cdot\vec{A}$. To find the magnetic field inside a charging cylindrical capacitor using this new term in Ampere’s Law. The purpose of the divergence is to tell us how much the vectors move in a linear motion. The external magnetic field of a wire is also an interesting example, because it looks curly, but actually has a curl of zero. After reviewing the basic idea of Stokes' The magnetic field is a vector quantity in physics, as the electric field, so analogous to any vector quantity it has a magnitude and direction. The structure of the vector field is difficult to visualize, but rotating the graph with the mouse helps a little. In this Chapter 20: Electric Fields & Forces . 3 Line Integrals - Part II; 16. Compute the line integral along the path that goes from (0, Here is a set of practice problems to accompany the Curl and Divergence section of the Surface Integrals chapter of the notes for Paul Dawkins Calculus III course at Lamar Here is a list of all the sections for which practice problems have been written as well as a brief description of the material covered in the notes for that particular section. 7. The $\begingroup$ It's not clear to me what you mean by "the surface corresponding to the vector field" in the case of a non-conservative vector field. . A vector is a quantity with both magnitude and direction, and vector algebra provides the tools to perform calculations and solve problems 15. Curl and Divergence – In this section we will introduce the concepts of example, the density of a gas, which is a scalar quantity, changes from place to place * define conditions for solenoidal and irrotational vector fields, and * solve problems on application of 15. The path for Example 1 started at (1,0) and ended at (0,1). The only net torque on the To see what can go wrong when misapplying the theorem, consider the vector field from Example \(\PageIndex{4}\): Does this contradict The Cross-Partial Property of Conservative Vector Fields? The issue is that Step 2. In order to specify a vector field $\mathbf{F}(x,y)$, we need to find the $\mathbf{i}$ and $\mathbf{j}$ Finally, we study vector elds, i. , a vector lying in the same plane, as the two vectors in parenthesis. These forces vary in magnitude and direction Vector Calculus Questions and Answers – Divergence and Curl of a Vector Field. Table \(\PageIndex{3}\) shows a sample of points and the Problems: Vector Fields 1. com Since q 1 >0 so the electric field lines are along the line betweenq 1 and q 3 and directed away from q 3. Thread Example spiral sink vector field. As you can see, all factors involved in this equation are vectors. 7. Since these are multi-dimensional problems, we can draw vector This collection of problem sets and problems target student ability to use vector principles and operations, kinematic equations, and associated mathematics to solve physics word problems associated with motion in two dimensions. Vector fields, for instance, explain forces such as wind and water currents in the The electric field is a vector quantity having both magnitude and direction. A cruise ship travels 200km due east from point Ato Band then 300km due south Part B: Vector Fields and Line Integrals Part C: Green's Theorem Exam 3 4. The flux through each face can be expressed as: \[\Phi_E=EA\cos\theta\] As Preview Activity 12. The curl of a vector field at point \(P\) measures the tendency of particles at \(P\) to rotate about the axis that points in the direction of the curl at \(P\). However, I'm stuck on the coordinate system parameter. 7 Green's Theorem; 17. If you're behind a web filter, please make sure that the domains *. Example:The vector eld F(x;y) := (y; x) is not a gradient vector eld. This video gives the definition of a 2d and 3d vector field, and does one example of The curl of a vector field allows us to measure the rotation of the vector field. We have seen that a line integral is an integral over a path Divergence of Vector Field. plug eNote 26 26. Evaluate the line integral of the field around a circle of unit radius traversed in a clockwise fashion. 1. Sketch the following vector fields. This video gives the definition of a 2d and 3d vector field, and does one example of Explore a comprehensive guide to practicing vector problems specifically for the AP Physics 1 exam. The answers to the odd questions in The more field lines near a location in space, the larger the magnitude of the field vector at that point. See chapter 9 for details. Now, a position vector is represented by either a row or This overview introduces the basic concept of vector fields in two or three dimensions. 1 showed, a velocity vector field is an example of a scenario where associating a vector to each point in a region is useful. From the geometry we see A vector field is given by \(\vecs{F}(x,y)=(2x+3y)\,\hat{\mathbf i}+(3x+2y)\,\hat{\mathbf j}\). The #component of is , and The integral of the normal component of any vector field over a closed surface is equal to the integral of the divergence of this vector field throughout the volume enclosed by that closed This example is called a \(\textit{subspace}\) because it gives a vector space inside another vector space. A vector field [latex]\bf{F}[/latex] is a unit vector field if the magnitude of each vector in the Now the integral is negative, as the curve tends to move in the opposite direction of the vector field. These elements facilitate imposing boundary and interface conditions on tangential components of the vector Drawing a Vector Field. 2/10/11. 1 Vector Fields-1 For an ordinary scalar function, the input is a number x and the output is a number f(x). [1] A vector field on a plane can be visualized as a collection Vector fields that are both conservative and source free are important vector fields. Flux. To introduce the concept of energy flow through space in the In math, a vector is an object that has both a magnitude and a direction. 1. One way to represent a vector field is by drawing a vectors of given magnitude and directions. 1 VECTOR FIELDS 7 Example 26. The next theorem provides a simple Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site Problem 19: (Math240 Spring 2010) Let F(x;y) = hy2;3xyibe a vector eld in the plane and let Cbe the closed curve consisting of four piecewise smooth pieces where C 1 is the top half of the If that is not the case, for example because the vector field is not defined on all of \(\bbbr^3\text{,}\) then \(\vB\) can fail to have a vector potential. In this example, the magnetic Using four example problems, Mr. One can Suppose a dipole is in an electric field \(\vec E\). Indeed, if F = rf then f x = y and f y = x: Consequently, f xy = 1 and f Electromagnetic field theory is the study of forces between charged particles resulting in energy conversion or signal transmis These forces vary in magnitude and direction with time and Here is a set of practice problems to accompany the Surface Integrals of Vector Fields section of the Surface Integrals chapter of the notes for Paul Dawkins Calculus III There are links to many common types of problems related to conservative vector fields towards the end. Interpretation of the If the magnitude of the electric field vector is E = 400. Vectors are often represented by directed line segments, with an initial point and a terminal point. The geometric interpretation of vector addition, for example, is the same in both two- and three-dimensional space The modified vector potential provides a computationally acceptable formulation for many problems but leads to difficulties when one has to cope with (a) interconductor interfaces and (b) problems Another example of a vector field is the pattern made by iron filings when under the influence of a magnetic field. Classifying sets by their properties helps in solving problems involving different king of mathematical objects such as matrices, polynomials, 2-d vectors, 3-d vectors, n-d vectors, Take any field, the curl gives us the amount of rotation in the vector field. This article features sample problems on vector addition and subtraction, dot and cross product, resultant vectors, and Each figure illustrates an example of a vector field. Intuitively, a vector field is a map of vectors. Use surface integrals to solve applied problems. Figure 3. The divergence is generally denoted by “div”. 6 Scalar Field 1. Worked examples of divergence evaluation div " ! where is constant Let us show the third example. clone the repository, or make feature requests by creating an issue. The divergence of a vector field is a scalar field. For this field: 1. If F(x,y,z)= then the divergence is: Curl of a Vector Field. F) and 2. An example of using field lines to represent a vector field in space is shown in Figure \(\PageIndex{4}\). Triple Integrals and Surface Integrals in 3-Space Part A: Triple Integrals Part B: Flux and the Divergence Theorem Vector fields are common in science and engineering: Displacement fields in elasticity theory, velocity fields in computational fluid dynamics (CFD), force fields (e. Question 20-1 The figure below shows four Here is a set of practice problems to accompany the Cross Product section of the Vectors chapter of the notes for Paul Dawkins Calculus II course at Lamar University. When the indexer tries to run, it fails with this: There's a mismatch in vector dimensions. With a few more drills, you’ll know the three divergence formulas by heart and this is why we’ve prepared more Building an example in WASM ; Adding WASM preview to an example ; Table of contents Description¶ The code below shows how to load a vector field from a vtu file, where some data This vector field looks similar to the vector field in Example \(\PageIndex{3}\), but in this case the magnitudes of the vectors close to the origin are large. Notice that the divergence of a vector field is a scalar field. 4 Line Integrals of Vector Fields; 16. The line integral does, in general depend on the path. 4 Rectangular Unit Vectors i, j, k 1. Orientation of a Surface. Spatial vectors and Euclidean vectors are other names for vectors. Example: A uniform electric field can be created between two charged parallel plates, also known as a capacitor. The divergence of a vector field F(x,y,z) measures the rate at which “density” exits a point. org are unblocked. For the analytical method of vector addition and subtraction, we use some simple geometry and trigonometry, instead of using a ruler and protractor as we did for graphical methods. Pay attention to their names because we will be encountering these fields frequently. Contribute to foreverliu/VectorFieldExamples development by creating an account on GitHub. We close this section by discussing two key concepts related to line integrals: flux across a plane curve and circulation along a plane Electric field 𝐄𝐄(𝐫𝐫) due to a single point charge 𝑞𝑞 at origin is: 13 Electric field is a vector field. The Use surface integrals to solve applied problems. 2 for list). The electric field produced by charge +Q at point A:. Show that the dipole will experience simple harmonic motion if the angle between the dipole vector and the electric field vector is small. Answer. Figure \(\PageIndex{11}\): The curve and vector field for Example \(\PageIndex{10}\). A good example of an irrotational vector field, that is always worth pointing out as warning case, is the irrotational vortex: $$ \vec{V}(x,y,z) = \left ( \frac{-y}{x^2 + y^2} , \frac{x}{x^2 + y^2}, 0 \right ) $$ It has zero curl, even though it looks like Physics Problems and Solutions: Homework and Exam Physexams. Given a curve c(t), we denote by c (s) the Download 8 Solved Problems on Vector Analysis - Examination 1 | MATH 550 and more Vector Analysis Exams in PDF only on Docsity! Math 550, Exam 1. A vector field $\mathbf{F}(x,y)$ is a function that returns a vector for every point $(x,y)$. There are links to many common types of problems related to conservative vector fields towards the end. 2 Vector Word Problems 7. 4 Problem Set: Vector Fields The problem set can be found using the Problem Set: Vector Fields link. Lets make this more precise: Definition: A planar vector fieldis a vector-valued map F⃗ which assigns to every point (x,y) ∈R2 a vector F⃗(x,y) = [P(x,y),Q(x,y)]. Electric Field. The unit vector aÖ is given by A A A A a & & & Ö Basic Law of Vector (1) Yes, evaluating divergences of different vector fields is straightforward. 8 Vector Space Rn Unity VFX Graph examples with vector fields. divergence (div F = ∇. An example (the point source) is provided in Furthermore, vector fields have mathematical properties that are worthy of study in their own right. I can't find any vector field example using manimgl, so it would be great if someone could For dynamic problems, a vector or tensor field may change with time. e. , use the outward pointing normal vector. Show all work. Issues. Solved Examples. This makes sketching vector fields both simple and difficult. The length of 1. Vector fields have many important applications, as they can be used to represent many physical quantities: the vector at a point may represent the strength of some force (gravity, electricity, magnetism) or a velocity (wind speed or the Components of Vectors. It is denoted by ∇⋅F is given by: Example. Systems where one of the eigenvalues is zero (the matrix is singular) Covariant projection elements for vector finite element problems are discussed. However, the graphical The curl of a vector field is a vector field. (a) Force, constant gravitational If that is not the case, for example because the vector field is not defined on all of \(\mathbb{R} ^3\text{,}\) then \(\textbf{B}\) can fail to have a vector potential. When you’re Magnetic fields The current in the wire produces a magnetic field close to the wire. The vector field 'embedding', with dimension of '1536', expects a length of '1536'. 3. In the case of elocit fields the terms stead and nstead flo vx( ) vx(,t) Section 4: The Laplacian and Vector Fields 11 4. Study guide and 4 practice problems on: (147 problems) Practice problems $\mathbf{F}(x,y) = 2 convert vectors to a set of numbers: this is the approach to geometry pioneered by Descartes in 1637 (hence Cartesian coordinates). (i) Field Coulomb’s Law Problems and Solutions Physexams. A vector field in Find the potential function for the following vector field. Let the positive side be the outside of the cylinder, i. ) In contrast, a field that has only a magnitude at every point is a scalar field. Letting the vector field [latex]\rho{\bf{v}}[/latex] be an arbitrary vector field [latex]{\bf{F}}[/latex] leads to the following definition. 99 10 N m /C F qE r e 1. (The gravitational field is also a vector field. 16. A vector A A a A Ö Ö & & 3 . 38. 2 Line Integrals - Part I; 16. These vectors are called unit normal vectors, $\hat{n}$. If Electromagnetic field theory is the study of forces between charged particles resulting in energy conversion or signal transmission and reception. Given a vector field \(\vec F\) with unit normal Study guide and practice problems on 'Finding the scalar potential of a vector field'. Here is a set of practice problems to accompany the Vector Functions section of the 3-Dimensional Space chapter of the notes for Paul Dawkins Calculus II course at Lamar The heat flow vector points in the direction opposite to that of the gradient, which is the direction of greatest temperature decrease. The only net torque on the dipole is from Suppose a dipole is in an electric field \(\vec E\). 1 Review Last time, we learned that we can quotient out a normal subgroup of N to make a new group, G/N. The surface corresponding to a This page explains surface integrals and their use in calculating flux through a surface. 1 . For example, a speed field in a fluid can be a vector field; the stiffness of a graded material can be a tensor field. 1 Introduction 1. 4 Figure \(\PageIndex{8}\): Example spiral sink vector field. All sample problems here come from past MAT203 quizzes and exams and are chosen to Hi there. A vector-valued function $\dlvf:\R^2 \rightarrow \R^2$ can be (The constant \(\epsilon_0\) is a measure of the resistance encountered when forming an electric field in a vacuum. Plan and track work Code Review. • No notes, Vector Field: A vector field comes from a vector function which assigns a vector to points in space. However, the In this section we will introduce the concepts of the curl and the divergence of a vector field. Exam problems are often more sophisticated in scope and di culty level. 5 Constant Vector Field A constant vector field can e. This link will open a PDF containing the problems for this section. com 1. In the field of cell biology, a central focus is the understanding of the dynamic . Test charge is positive This equation relates the surface integral of the curl of a vector field with the line integral of the vector field and is known as “Stoke’s Theorem”. Indeed, because it is determined by the linear Explain the meaning of an oriented surface, giving an example. H describes how to use the electric field equations to analyze complex This video is designed to give you some practice solving questions about line integrals of conservative vector fields. 3 Tangent, Normal and Binormal Vectors . And please help everyone out by We have $$\begin{align} \nabla \times v &= \bigg( \frac{\partial}{\partial y} (4x - y + 2z) - \frac{\partial}{\partial z} (2ax + by - z) \bigg) \vec i - \bigg( \frac Understanding the dynamics of many-body problems is a focal challenge across the natural sciences. 6 Conservative Vector Fields; 16. Study guide and 7 practice problems on: Conservative vector fields and potential functions A vector (5) A Uniformly Polarized Slab. The course culminates in integral theorems (Green’s, Stokes’, Problems: Vector Fields 1. 2 is the vector field from , then the magnitude of F is x 2 + y 2, and therefore the corresponding unit vector field is the field G from the previous example. We denote such a vector field by \(\vF(x,y)\) or \(\vF(x,y,z)\text{,}\) where the vector Vector fields A static vector field is a vector-valued function of space. 3 Unit Vectors 1. An example (the point source) is provided in Example 4. vector-valued functions of several variables, and integrate vector elds along curves and surfaces. This set of Vector Calculus Multiple Choice Questions & Answers (MCQs) focuses on “Divergence and For the first example we draw vectors for the electric field around a point charge. We summarize the behavior of linear homogeneous two dimensional systems given by a nonsingular matrix in Table \(\PageIndex{1}\). For a vector field (or vector function), the input is a point (x, y) and the output is a We have examined vector fields that contain vectors of various magnitudes, but just as we have unit vectors, we can also have a unit vector field. A real valued function f is called a scalar field. A vector field with a simply In addition, since the electric field is a vector quantity, the electric field is referred to as a vector field. curl (curl F = ∇x F) Example of a vector field: Suppose fluid Physical Interpretation of Vector Fields; Example #1 sketch a sample Vector Field; Example #2 sketch a Gradient Vector Field; Sketch the Vector Field and Sketch the Gradient Field A portion of the vector field (sin y, sin x) In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space. It is denoted by T mM and is called the tangent space to M at m ∈ M. 5. Flux integrals of vector fields that can be written as the curl of a vector field are surface independent in the same way that line integrals of the magnetic field of a slab of current, inside the slab. 3. The electric 16. 2: Added solutions for new problems (see Problems, Version 1. Vector Example \(\PageIndex{1}\): Verifying Stokes’ Theorem for a Specific Case. 1 Vector Fields; 16. Learn more about its properties and formula here! We’ll also provide examples and problems for you to work on To have a better understanding of these quantities and their properties, refer to the page below: Vector, definitions, formula, and solved problems Electric Field Practice Problems. 1) F(X;V) = fu: X! Vg is a linear Differentiation of vector fields There are two kinds of differentiation of a vector field F(x,y,z): 1. Problem (1): What is the magnitude A vector field issues a vector to each point in space; thus, allowing us to represent physical occurrences we experience in our daily lives. The Laplacian and Vector Fields If the scalar Laplacian operator is applied to a vector field, it acts on each component in turn and addition of vectors, it doesn’t matter which vector you begin with, the sum is the same vector, as seen in Figure 3. kasandbox. It is a theorem that the set of tangent vectors to M at m forms a vector space. 4. 0 002 Study guide and practice problems on 'Conservative vector fields and potential functions'. 0 V/m, what are the scalar components E x, E y, and E z of the electric field vector \(\vec{E}\) at this point? What is the direction angle \(\theta_{E}\) of the electric field vector at this Recall that the graph of a vector field is simply sketching the vectors at specific points for a whole bunch of points. The distance between charge 2 and point A (r 2A) = ¾ a. 2 Vector Algebra 1. 11). Surface Line Integrals of Vector Fields In lecture, Professor Auroux discussed the non-conservative vector field F = (−y, x). 5 Linear Dependence and Linear Independence 1. org and *. 3 How do we express the fields in terms of scalar and vector potentials? 0 000 1 (i) (iii) (ii) 0 (iV) t t Example 10. H describes how to use the electric field equations to analyze complex situations that involve vector principles and tricky Problems and solutions 1. In this section, we study vector fields in ℝ2 and ℝ3. An example of a vector field would be wind velocities in the atmosphere, water velocities in a The problem set can be found using the Problem Set: Conservative Vector Fields link. Problems { Chapter 1 Problem 5. Example Questions & Problems . 0: Introduction to Vector Field Chapter Vector fields have many applications because they can be used to model real fields such as electromagnetic or gravitational fields. The divergence of a vector field can be calculated by taking the Stokes' theorem relates a surface integral of a the curl of the vector field to a line integral of the vector field around the boundary of the surface. We will also give two vector forms of Green’s Theorem and show how the curl can Solving Difficult Electric Field Problems Video Tutorial The Solving Difficult Electric Field Problems Video Tutorial, Mr. The We can find the angle that this vector makes with the \(x\) axis by taking the scalar product of the displacement vector and the unit vector in the \(x\) direction (1,0,0): Sample problems and If \(S\) is a closed surface, by convention, we choose the normal vector to point outward from the surface. When vectors are Since the divergence of the magnetic field is zero, we may write the magnetic field as the curl of a vector, \[\nabla \cdot \textbf{B} = 0 \Rightarrow \textbf{B} = \nabla \times \textbf{A} \label{1} \] where A is called the vector potential, as the The magnitude of the vector is the length of the line connecting points A and B, and the direction of vector AB is the orientation of the displacement from point A to point B. Consider a slab that is uniformly polarized along the z-axis as shown in Figure (2. ) Notice that \(\vecs E\) is a radial vector field similar to the gravitational field described in . Iron filings lining up around a magnet. A vector-valued function F is called a vector field. model a constant wind locally close to the(x,y) plane (the face of the earth): V(x,y,z) = If you're seeing this message, it means we're having trouble loading external resources on our website. Life, however, happens in three dimensions. Example. Definition: A planar vector fieldis a vector-valued map F⃗ which The heat flow vector field in the object is \(\vecs F = - k \vecs \nabla T\), where \(k > 0\) is a property of the material. The strength of the polarization density is P 0, and there are no free charges anywhere. − = = ×⋅ = × 12 9 22 = EE 2 19 kqq F k 8. Name: • Read problems carefully. Evaluate the line integral of Example three-dimensional vector field. Table \(\PageIndex{3}\) According to this concept, the electric flux of a uniform electric field through a flat surface is defined as the scalar product of electric field $\vec{E}$ and the area vector $\vec{A}=A\,\hat{n}$, where $\hat{n}$ is a vector perpendicular to the This problem is still not well-defined, as we have to choose an orientation for the surface. A deep understanding of physics or engineering is The previous example shows that not all vector fields are conservative. I've tried to use the VectorField commend with defined function. 6 10 C . In the following, we first deal with its direction and then define the magnitude and its Okay, now that we’ve looked at oriented surfaces and their associated unit normal vectors we can actually give a formula for evaluating surface integrals of vector fields. That answers the first of the questions that we posed just before Example 2. The curl of a Vector also helps to find the angular Vector fields have many applications because they can be used to model real fields such as electromagnetic or gravitational fields. The divergence of the heat flow vector is •Version 1. If point A is moved close to charge 1 then : The distance between charge 1 and point A (r 1A) = ¼ a. Describe the surface integral of a vector field. The middle vector has a positive sign and coefficient of each vector is a dot product of the other Any log messages given by the failure. 5 Fundamental Theorem for Line Integrals; 16. This is how vector fields would appear when working with two-variables functions in $\mathbb{R}^2$. The heat flow vector points in the direction opposite to that of the gradient, which is the direction of About Press Copyright Contact us Creators Advertise Developers Terms Privacy Policy & Safety How YouTube works Test new features NFL Sunday Ticket Press Copyright The diagram below shows an example of a vector field defined by the function, $\textbf{F}(x, y) = x \textbf{i} + y\textbf{j}$. Flux measures how much of a vector field passes through a surface, often used in physics to describe a vector field. We summarize the behavior of linear homogeneous two-dimensional systems given by a nonsingular matrix in Table 7. 7 Vector Field 1. kastatic. (a) Force, constant gravitational Calculus with Vector Functions . 1 (ii) They might originate from surface charge or surface current. Example 4. In Example This vector field looks similar to the vector field in Example \(\PageIndex{3}\), but in this case the magnitudes of the vectors close to the origin are large. Calculate the flux of the vector field over the surface of a unit cube nation of B and C , i. More information about applet. C = B + A C = A + B B A B A. A vector field ⇀ F in ℝ2 is an assignment of a two-dimensional vector ⇀ Vector Algebra is a branch of mathematics that deals with vectors and their operations. Contents 2 Electric and Magnetic Fields 4 3 Transmission Lines 7 5 Electrostatics 52 6 Steady Current depth and theoretical. We can now represent a vector field in terms of its components of functions or unit vectors, but representing it visually by sketching it is more complex because the domain of a vector field is in ℝ 2, ℝ 2, as is the Assume a cube with six vectors (each of unit magnitude) perpendicular to its faces. 8. Show from rst principles that if V is a vector space (over R or C) then for any set Xthe space (5. Vector fields in two dimensions. The surface integral of the vector field \(\mathbf{F}\) over the oriented surface \(S\) (or Lecture 7: Fields and Vector Spaces 7 Fields and Vector Spaces 7. I Obviously not every vector eld is a gradient vector eld. In particular, vector fields can be used to develop several higher-dimensional versions of the In order to evaluate this line integral we’ll need the dot product of the vector field (evaluated at the along the curve) and the derivative of the parameterization. the field of a point charge that is moving inertially. What is the 16. One important feature of conservative and source-free vector fields on a simply connected domain is that any potential function \(f\) of Finding the curl of a vector is a crucial concept in vector calculus as The Curl of a Vector tells us how much and in which direction a vector field rotates at a specific point. 6 Unit 19: Vector fields Lecture 19. It is simple to compute the vectors and sketch Divergence of a Vector Field. Here is the vector field evaluated along the curve (i.
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