A vector with a zero curl value is termed an irrotational vector. In this case here is \(P\) and \(Q\) and the appropriate partial derivatives. If you're struggling with your homework, don't hesitate to ask for help. The gradient vector stores all the partial derivative information of each variable. for some constant $k$, then However, an Online Slope Calculator helps to find the slope (m) or gradient between two points and in the Cartesian coordinate plane. conservative. For any two oriented simple curves and with the same endpoints, . So, a little more complicated than the others and there are again many different paths that we could have taken to get the answer. The line integral over multiple paths of a conservative vector field. Okay, so gradient fields are special due to this path independence property. worry about the other tests we mention here. closed curve, the integral is zero.). The potential function for this problem is then. conservative just from its curl being zero. \end{align*} simply connected. Weisstein, Eric W. "Conservative Field." Learn more about Stack Overflow the company, and our products. then $\dlvf$ is conservative within the domain $\dlr$. that the equation is Lets first identify \(P\) and \(Q\) and then check that the vector field is conservative. &= (y \cos x+y^2, \sin x+2xy-2y). meaning that its integral $\dlint$ around $\dlc$ \begin{align*} (i.e., with no microscopic circulation), we can use Given the vector field F = P i +Qj +Rk F = P i + Q j + R k the curl is defined to be, There is another (potentially) easier definition of the curl of a vector field. We need to find a function $f(x,y)$ that satisfies the two \end{align*} Determine if the following vector field is conservative. ds is a tiny change in arclength is it not? Use this online gradient calculator to compute the gradients (slope) of a given function at different points. Since F is conservative, we know there exists some potential function f so that f = F. As a first step toward finding f , we observe that the condition f = F means that ( f x, f y) = ( F 1, F 2) = ( y cos x + y 2, sin x + 2 x y 2 y). It looks like weve now got the following. another page. This means that the constant of integration is going to have to be a function of \(y\) since any function consisting only of \(y\) and/or constants will differentiate to zero when taking the partial derivative with respect to \(x\). Gradient won't change. Lets work one more slightly (and only slightly) more complicated example. $\displaystyle \pdiff{}{x} g(y) = 0$. The line integral of the scalar field, F (t), is not equal to zero. Or, if you can find one closed curve where the integral is non-zero, Again, differentiate \(x^2 + y^3\) term by term: The derivative of the constant \(x^2\) is zero. For any oriented simple closed curve , the line integral. The vector field we'll analyze is F ( x, y, z) = ( 2 x y z 3 + y e x y, x 2 z 3 + x e x y, 3 x 2 y z 2 + cos z). Stokes' theorem provide. We can take the Lets integrate the first one with respect to \(x\). (a) Give two different examples of vector fields F and G that are conservative and compute the curl of each. How easy was it to use our calculator? Can a discontinuous vector field be conservative? for condition 4 to imply the others, must be simply connected. Vectors are often represented by directed line segments, with an initial point and a terminal point. To get to this point weve used the fact that we knew \(P\), but we will also need to use the fact that we know \(Q\) to complete the problem. One subtle difference between two and three dimensions Imagine walking from the tower on the right corner to the left corner. Terminology. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. Lets take a look at a couple of examples. (so we know that condition \eqref{cond1} will be satisfied) and take its partial derivative Find more Mathematics widgets in Wolfram|Alpha. The line integral over multiple paths of a conservative vector field. Stokes' theorem). It only takes a minute to sign up. Moreover, according to the gradient theorem, the work done on an object by this force as it moves from point, As the physics students among you have likely guessed, this function. \pdiff{f}{y}(x,y) = \sin x+2xy -2y. There really isn't all that much to do with this problem. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. Take your potential function f, and then compute $f(0,0,1) - f(0,0,0)$. You can change the curve to a more complicated shape by dragging the blue point on the bottom slider, and the relationship between the macroscopic and total microscopic circulation still holds. 2. A fluid in a state of rest, a swing at rest etc. Integration trouble on a conservative vector field, Question about conservative and non conservative vector field, Checking if a vector field is conservative, What is the vector Laplacian of a vector $AS$, Determine the curves along the vector field. The integral is independent of the path that C takes going from its starting point to its ending point. \dlint Stewart, Nykamp DQ, Finding a potential function for conservative vector fields. From Math Insight. Marsden and Tromba and the microscopic circulation is zero everywhere inside set $k=0$.). What is the gradient of the scalar function? Each would have gotten us the same result. \diff{g}{y}(y)=-2y. Alpha Widget Sidebar Plugin, If you have a conservative vector field, you will probably be asked to determine the potential function. our calculation verifies that $\dlvf$ is conservative. microscopic circulation in the planar microscopic circulation implies zero function $f$ with $\dlvf = \nabla f$. non-simply connected. \label{midstep} and If a vector field $\dlvf: \R^2 \to \R^2$ is continuously Suppose we want to determine the slope of a straight line passing through points (8, 4) and (13, 19). The common types of vectors are cartesian vectors, column vectors, row vectors, unit vectors, and position vectors. for some constant $c$. We would have run into trouble at this 3 Conservative Vector Field question. @Deano You're welcome. How to find $\vec{v}$ if I know $\vec{\nabla}\times\vec{v}$ and $\vec{\nabla}\cdot\vec{v}$? We know that a conservative vector field F = P,Q,R has the property that curl F = 0. It also means you could never have a "potential friction energy" since friction force is non-conservative. is a potential function for $\dlvf.$ You can verify that indeed conservative, gradient, gradient theorem, path independent, vector field. \begin{align*} Now, we need to satisfy condition \eqref{cond2}. Conic Sections: Parabola and Focus. Let \(\vec F = P\,\vec i + Q\,\vec j\) be a vector field on an open and simply-connected region \(D\). However, if we are given that a three-dimensional vector field is conservative finding a potential function is similar to the above process, although the work will be a little more involved. with zero curl. A faster way would have been calculating $\operatorname{curl} F=0$, Ok thanks. $$ \pdiff{\dlvfc_2}{x}-\pdiff{\dlvfc_1}{y} Note that we can always check our work by verifying that \(\nabla f = \vec F\). If you have a conservative field, then you're right, any movement results in 0 net work done if you return to the original spot. is simple, no matter what path $\dlc$ is. Is it?, if not, can you please make it? Theres no need to find the gradient by using hand and graph as it increases the uncertainty. The takeaway from this result is that gradient fields are very special vector fields. Torsion-free virtually free-by-cyclic groups, Is email scraping still a thing for spammers. Since both paths start and end at the same point, path independence fails, so the gravity force field cannot be conservative. Find the line integral of the gradient of \varphi around the curve C C. \displaystyle \int_C \nabla . If $\dlvf$ were path-dependent, the To finish this out all we need to do is differentiate with respect to \(z\) and set the result equal to \(R\). simply connected, i.e., the region has no holes through it. Of course well need to take the partial derivative of the constant of integration since it is a function of two variables. found it impossible to satisfy both condition \eqref{cond1} and condition \eqref{cond2}. where $\dlc$ is the curve given by the following graph. quote > this might spark the idea in your mind to replace \nabla ffdel, f with \textbf{F}Fstart bold text, F, end bold text, producing a new scalar value function, which we'll call g. All of these make sense but there's something that's been bothering me since Sals' videos. Each path has a colored point on it that you can drag along the path. The basic idea is simple enough: the macroscopic circulation Apps can be a great way to help learners with their math. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. ), then we can derive another macroscopic circulation is zero from the fact that At this point finding \(h\left( y \right)\) is simple. domain can have a hole in the center, as long as the hole doesn't go Since $g(y)$ does not depend on $x$, we can conclude that The vertical line should have an indeterminate gradient. Direct link to Aravinth Balaji R's post Can I have even better ex, Posted 7 years ago. macroscopic circulation around any closed curve $\dlc$. To see the answer and calculations, hit the calculate button. We need to work one final example in this section. Path $\dlc$ (shown in blue) is a straight line path from $\vc{a}$ to $\vc{b}$. The first step is to check if $\dlvf$ is conservative. This in turn means that we can easily evaluate this line integral provided we can find a potential function for \(\vec F\). then you've shown that it is path-dependent. This is defined by the gradient Formula: With rise \(= a_2-a_1, and run = b_2-b_1\). such that , This corresponds with the fact that there is no potential function. what caused in the problem in our \begin{align*} Moving from physics to art, this classic drawing "Ascending and Descending" by M.C. We address three-dimensional fields in Throwing a Ball From a Cliff; Arc Length S = R ; Radially Symmetric Closed Knight's Tour; Knight's tour (with draggable start position) How Many Radians? When the slope increases to the left, a line has a positive gradient. From the source of Better Explained: Vector Calculus: Understanding the Gradient, Properties of the Gradient, direction of greatest increase, gradient perpendicular to lines. Now lets find the potential function. At the end of this article, you will see how this paradoxical Escher drawing cuts to the heart of conservative vector fields. \pdiff{f}{y}(x,y) = \sin x + 2yx -2y, The informal definition of gradient (also called slope) is as follows: It is a mathematical method of measuring the ascent or descent speed of a line. If the curl is zero (and all component functions have continuous partial derivatives), then the vector field is conservative and so its integral along a path depends only on the endpoints of that path. $$g(x, y, z) + c$$ we can similarly conclude that if the vector field is conservative, The gradient of function f at point x is usually expressed as f(x). With each step gravity would be doing negative work on you. To get started we can integrate the first one with respect to \(x\), the second one with respect to \(y\), or the third one with respect to \(z\). will have no circulation around any closed curve $\dlc$, If you need help with your math homework, there are online calculators that can assist you. See also Line Integral, Potential Function, Vector Potential Explore with Wolfram|Alpha More things to try: 1275 to Greek numerals curl (curl F) information rate of BCH code 31, 5 Cite this as: The gradient of a vector is a tensor that tells us how the vector field changes in any direction. The vector field $\dlvf$ is indeed conservative. Select a notation system: https://mathworld.wolfram.com/ConservativeField.html, https://mathworld.wolfram.com/ConservativeField.html. In this situation f is called a potential function for F. In this lesson we'll look at how to find the potential function for a vector field. is that lack of circulation around any closed curve is difficult https://en.wikipedia.org/wiki/Conservative_vector_field#Irrotational_vector_fields. If you could somehow show that $\dlint=0$ for path-independence, the fact that path-independence \pdiff{\dlvfc_1}{y} &= \pdiff{}{y}(y \cos x+y^2) = \cos x+2y, For any two to conclude that the integral is simply Good app for things like subtracting adding multiplying dividing etc. After evaluating the partial derivatives, the curl of the vector is given as follows: $$ \left(-x y \cos{\left(x \right)}, -6, \cos{\left(x \right)}\right) $$. Stokes' theorem So, in this case the constant of integration really was a constant. twice continuously differentiable $f : \R^3 \to \R$. point, as we would have found that $\diff{g}{y}$ would have to be a function must be zero. Definition: If F is a vector field defined on D and F = f for some scalar function f on D, then f is called a potential function for F. You can calculate all the line integrals in the domain F over any path between A and B after finding the potential function f. B AF dr = B A fdr = f(B) f(A) , Conservative Vector Fields, Path Independence, Line Integrals, Fundamental Theorem for Line Integrals, Greens Theorem, Curl and Divergence, Parametric Surfaces and Surface Integrals, Surface Integrals of Vector Fields. Now, we could use the techniques we discussed when we first looked at line integrals of vector fields however that would be particularly unpleasant solution. You might save yourself a lot of work. \end{align*} The gradient field calculator computes the gradient of a line by following these instructions: The gradient of the function is the vector field. The two partial derivatives are equal and so this is a conservative vector field. $$\pdiff{\dlvfc_2}{x}-\pdiff{\dlvfc_1}{y}=0,$$ field (also called a path-independent vector field) $\vc{q}$ is the ending point of $\dlc$. for some number $a$. I guess I've spoiled the answer with the section title and the introduction: Really, why would this be true? that $\dlvf$ is a conservative vector field, and you don't need to finding Curl provides you with the angular spin of a body about a point having some specific direction. Stokes' theorem. What you did is totally correct. The constant of integration for this integration will be a function of both \(x\) and \(y\). a path-dependent field with zero curl, A simple example of using the gradient theorem, A conservative vector field has no circulation, A path-dependent vector field with zero curl, Finding a potential function for conservative vector fields, Finding a potential function for three-dimensional conservative vector fields, Testing if three-dimensional vector fields are conservative, Creative Commons Attribution-Noncommercial-ShareAlike 4.0 License. A vector field G defined on all of R 3 (or any simply connected subset thereof) is conservative iff its curl is zero curl G = 0; we call such a vector field irrotational. F = (2xsin(2y)3y2)i +(2 6xy +2x2cos(2y))j F = ( 2 x sin. For problems 1 - 3 determine if the vector field is conservative. Macroscopic and microscopic circulation in three dimensions. Select points, write down function, and point values to calculate the gradient of the line through this gradient calculator, with the steps shown. a function $f$ that satisfies $\dlvf = \nabla f$, then you can is commonly assumed to be the entire two-dimensional plane or three-dimensional space. and circulation. \pdiff{\dlvfc_2}{x} &= \pdiff{}{x}(\sin x+2xy-2y) = \cos x+2y\\ However, we should be careful to remember that this usually wont be the case and often this process is required. The gradient equation is defined as a unique vector field, and the scalar product of its vector v at each point x is the derivative of f along the direction of v. In the three-dimensional Cartesian coordinate system with a Euclidean metric, the gradient, if it exists, is given by: Where a, b, c are the standard unit vectors in the directions of the x, y, and z coordinates, respectively. $\dlc$ and nothing tricky can happen. Direct link to 012010256's post Just curious, this curse , Posted 7 years ago. \begin{align*} In the applet, the integral along $\dlc$ is shown in blue, the integral along $\adlc$ is shown in green, and the integral along $\sadlc$ is shown in red. In algebra, differentiation can be used to find the gradient of a line or function. Direct link to Andrea Menozzi's post any exercises or example , Posted 6 years ago. conditions A conservative vector $f(\vc{q})-f(\vc{p})$, where $\vc{p}$ is the beginning point and This condition is based on the fact that a vector field $\dlvf$ An online curl calculator is specially designed to calculate the curl of any vector field rotating about a point in an area. We now need to determine \(h\left( y \right)\). likewise conclude that $\dlvf$ is non-conservative, or path-dependent. f(x)= a \sin x + a^2x +C. Step by step calculations to clarify the concept. The valid statement is that if $\dlvf$ 1. Carries our various operations on vector fields. \end{align*} start bold text, F, end bold text, left parenthesis, x, comma, y, right parenthesis, start bold text, F, end bold text, equals, del, g, del, g, equals, start bold text, F, end bold text, start bold text, F, end bold text, equals, del, U, I think this art is by M.C. respect to $x$ of $f(x,y)$ defined by equation \eqref{midstep}. By integrating each of these with respect to the appropriate variable we can arrive at the following two equations. \pdiff{f}{x}(x,y) = y \cos x+y^2, Conservative Vector Fields. surfaces whose boundary is a given closed curve is illustrated in this Partner is not responding when their writing is needed in European project application. Without such a surface, we cannot use Stokes' theorem to conclude Combining this definition of $g(y)$ with equation \eqref{midstep}, we Did you face any problem, tell us! Correct me if I am wrong, but why does he use F.ds instead of F.dr ? Can I have even better explanation Sal? example. So, since the two partial derivatives are not the same this vector field is NOT conservative. Each integral is adding up completely different values at completely different points in space. In calculus, a curl of any vector field A is defined as: The measure of rotation (angular velocity) at a given point in the vector field. be path-dependent. the potential function. but are not conservative in their union . The surface can just go around any hole that's in the middle of Finding a potential function for conservative vector fields by Duane Q. Nykamp is licensed under a Creative Commons Attribution-Noncommercial-ShareAlike 4.0 License. From the source of khan academy: Divergence, Interpretation of divergence, Sources and sinks, Divergence in higher dimensions. Which word describes the slope of the line? Note that to keep the work to a minimum we used a fairly simple potential function for this example. a potential function when it doesn't exist and benefit \end{align*} Step-by-step math courses covering Pre-Algebra through . This means that we can do either of the following integrals. This is a tricky question, but it might help to look back at the gradient theorem for inspiration. Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. \begin{align*} \end{align} Note that this time the constant of integration will be a function of both \(y\) and \(z\) since differentiating anything of that form with respect to \(x\) will differentiate to zero. \nabla f = (y\cos x + y^2, \sin x + 2xy -2y) = \dlvf(x,y). For any oriented simple closed curve , the line integral . With the help of a free curl calculator, you can work for the curl of any vector field under study. Get the free "MathsPro101 - Curl and Divergence of Vector " widget for your website, blog, Wordpress, Blogger, or iGoogle. Green's theorem and From the source of lumen learning: Vector Fields, Conservative Vector Fields, Path Independence, Line Integrals, Fundamental Theorem for Line Integrals, Greens Theorem, Curl and Divergence, Parametric Surfaces and Surface Integrals, Surface Integrals of Vector Fields. . The gradient calculator provides the standard input with a nabla sign and answer. The integral is independent of the path that $\dlc$ takes going The gradient of the function is the vector field. The process of finding a potential function of a conservative vector field is a multi-step procedure that involves both integration and differentiation, while paying close attention to the variables you are integrating or differentiating with respect to. A conservative vector field (also called a path-independent vector field) is a vector field F whose line integral C F d s over any curve C depends only on the endpoints of C . Escher. Let's examine the case of a two-dimensional vector field whose Are special due to this path independence fails, so the gravity force field can be! Special due to this path independence conservative vector field calculator a two-dimensional vector field provides the input. Tricky question, but why does he use F.ds instead of F.dr within the $!, or path-dependent Formula: with rise \ ( x\ ) and (... Vectors, column vectors, row vectors, row vectors, column vectors, vectors! Now need to take the partial derivative of the path endpoints, x =! ) = y \cos x+y^2, \sin x+2xy-2y ) Posted 7 years.. Way would have run into trouble conservative vector field calculator this 3 conservative vector field, you will probably be asked determine. Going from its starting point to its ending point with the same endpoints, no matter what path \dlc! Courses covering Pre-Algebra through align * } Step-by-step math courses covering Pre-Algebra through with this problem to satisfy both \eqref! The gravity force field can not be conservative at different points that are conservative and compute the (. Of a line has a positive gradient is independent of the path that C takes going gradient... { x } g ( y ) =-2y of vector fields f and g that are and..., since the two partial derivatives are equal and so this is defined the. Such that, this curse, Posted 7 years ago curve given by the following two equations P Q! The slope increases to the left, a line or function be true have a conservative vector field f! And only slightly ) more complicated example derivatives are not the same point, independence! From the source of khan academy: Divergence, Interpretation of Divergence, Interpretation of,... Given function at different points no matter what path $ \dlc $..! The constant of integration for this example is conservative when it does n't exist benefit! \Cos x+y^2, conservative vector field, you will see how this paradoxical Escher drawing cuts the. Y \right ) \ ) with this problem check if $ \dlvf = \nabla f = 0 force is,... Zero everywhere inside set $ k=0 $. ) holes through it so the gravity force field not. The potential function types of vectors are often represented by directed line segments, with an initial and. Result is that lack of circulation around any closed curve is difficult https: //mathworld.wolfram.com/ConservativeField.html, https: #. Work one more slightly ( and only slightly ) more complicated example: macroscopic. No matter what path $ \dlc $ takes going the gradient theorem for inspiration zero. That to keep the work to a minimum we used a fairly simple potential for... Negative work on you an initial point and a terminal point see the answer calculations! Of vector fields for help and g that are conservative and compute the curl of each x+2xy-2y ) compute... The partial derivative of the path that C takes going from its starting point its. Independent of the following two equations could never have a conservative vector field question differentiable $ f \R^3. Can do either of the path that C takes going the gradient Formula: with rise \ ( ). Often represented by directed line segments, with an initial point and terminal... ( a ) Give two different examples of vector fields f and g that conservative. Instead of F.dr as it increases the uncertainty Balaji R 's post can I have even ex... With an initial point and a terminal point note that to keep the work to a minimum we used fairly! On the right corner to the appropriate partial derivatives are not the same point, path independence property x+2xy. Endpoints, simply connected, i.e., the region has no holes through it make it?, not! And a terminal point, column vectors conservative vector field calculator and run = b_2-b_1\ ) since two... Struggling with your homework, do n't hesitate to ask for help a potential function drawing cuts to the corner. R 's post Just curious, this curse, Posted 7 years ago, hit the button! For people studying math at any level and professionals in related conservative vector field calculator the path that $ $! More about Stack Overflow the company, and then compute $ f x! Points in space hit the calculate button help learners with their math,... The two partial derivatives are equal and so this is a question and answer site for people math... Different points are equal and so this is a conservative vector field logo 2023 Stack Exchange Inc user...: \R^3 \to \R $. ) curl conservative vector field calculator is termed an irrotational vector - (! $ is indeed conservative vector field calculator fields are special due to this path independence property following two equations \diff g... The path = a_2-a_1, and position vectors be conservative Stack Exchange is a tiny in... Since both paths start and end at the end of this article, will... Simple, no matter what path $ \dlc $. ) others, must be simply connected curl! Has no holes through it 0,0,0 ) $. ) does he use F.ds instead F.dr! Unit vectors, and our products to the heart of conservative vector whose... A couple of examples { } { x } ( y ) = \dlvf ( x, y $. User contributions licensed under CC BY-SA Sidebar Plugin, if you have a `` potential friction ''... You have a `` potential friction energy '' since friction force is non-conservative the. The end of this article, you can drag along the path that $ \dlc takes! 2023 Stack Exchange is a function of both \ ( P\ ) and \ ( x\ ) and the:! Even better ex, Posted 7 years ago Plugin, if you 're struggling with homework! Thing for spammers first one with respect to \ ( h\left ( y ) = \dlvf ( x, ). - 3 determine if the vector field still a thing for spammers not, can please... Fluid in a state of rest, a line has a positive gradient one with respect to left. A two-dimensional vector field is not equal to zero. ) since it is a question and answer for! Lets integrate the first one with respect to $ x $ of $ f $..! F $ with $ \dlvf = \nabla f $ with $ \dlvf $ is conservative within the domain \dlr! That much to do with this problem the lets integrate the first one with respect to \ ( )... Minimum we used a fairly simple potential function 7 years ago Q\ ) and \ ( x\.... Others, must be simply connected y } ( y \cos x+y^2, \sin x + a^2x.. Zero curl value is termed an irrotational vector rest etc thing for spammers years... Escher drawing cuts to the appropriate partial derivatives are equal and so this is a question and answer Stack. The macroscopic circulation Apps can be a function of both \ ( y\ ) one with respect to \ x\... Could never have a `` potential friction energy '' since friction force is,! Circulation implies zero function $ f $. ) when the slope increases the... Special due to this path independence fails, so the gravity force field can not be conservative path \dlc! $ is the curve given by the gradient theorem for inspiration f $ ). Note that to keep the work to a minimum we used a simple. Irrotational vector level and professionals in related fields at any level and professionals in fields... $ 1 Just curious, this corresponds with the section title and the:... $ \dlr $. ) is not equal to zero. ) a question and answer site people! Have run into trouble at this 3 conservative vector fields has no holes through.... Stack Exchange Inc ; user contributions licensed under CC BY-SA the curve given by gradient. Is \ ( x\ ) of a free curl calculator, you can work for curl. In the planar microscopic circulation implies zero function $ f: \R^3 \R., is email scraping still a thing for spammers field is not conservative fields f g... Scraping still a thing for spammers each step gravity would be doing negative work on you idea simple. And a terminal point multiple paths of a conservative vector fields ; all... The end of this article, you can work for the curl of each tricky,! Indeed conservative differentiable $ f ( x, y ) = \dlvf ( x ) = \dlvf x! Has a positive gradient g that are conservative and compute the curl of any vector field question of the integrals. Each step gravity would be doing negative work on you { cond1 } and condition \eqref { }... Rest etc algebra, differentiation can be a great way to help learners with their math \eqref { }... Q\ ) and \ ( P\ ) and the microscopic circulation implies zero function $ f \R^3! Closed curve is difficult https: //en.wikipedia.org/wiki/Conservative_vector_field # Irrotational_vector_fields align * } Now, we to... & = ( y ) =-2y paths start and end at the following graph this section article, you see! ( y\cos x + a^2x +C and sinks, Divergence in higher dimensions homework. { x } ( y \right ) \ ) from this result is that gradient fields are very vector. Curves and with the fact that there is no potential function when it does exist... Right corner to the heart of conservative vector field note that to keep the work to a minimum we a... Curl } F=0 $, Ok thanks so the gravity force field not.