Wikipedia:Reference desk/Archives/Mathematics/2011 September 12

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September 12

Checking Conservativeness

Since we know that the curl of a conservative vector field is always zero, we can use this fact to check if a given field is conservative or not (without actually finding the potential). The Curl operator works only on 3D fields (no more, no less). So for a 2D field, we add the third component as being zero and then check its curl. For a 3D field, we have no problem. My question is what can I do for a higher dimensional field? How can I check to see if its conservative without actually finding the potential? Is there a shortcut? I am thinking maybe I can just generalize from the 2D-3D case and check all the partials to see if they match but would that work? If yes, how can you justify this method? Would this still be an if and only if relationship in higher dimensions? Thanks!67.40.134.8 (talk) 02:42, 12 September 2011 (UTC)

A conservative field is one which the gradient of a function. In dimension 3, the curl being 0 states that all the mixed partial derivatives of the function are equal. In higher dimensions you can get the same effect by computing the wedge product ∇∧f where f is the field. This computes the differences between the partial derivatives (there are 6 in dimension 4, 10 in dimension 5, ...) and if the differences are all 0 then the field is conservative, though you need a higher dimensional version of Stokes' theorem to make that rigorous.--RDBury (talk) 15:49, 12 September 2011 (UTC)

What you said is nearly right, except the topology of the domain is important. For instance, the curl of the vector field $-y/(x^{2}+y^{2})i+x/(x^{2}+y^{2})j$ is zero, but this vector field is not conservative since its line integral over a loop enclosing the origin is nonzero. A correct statement would be that a vector field in a plane domain that is simply connected (having no holes) is conservative if and only if its curl is zero. However, even with this criterion here you need to be careful, since there are vector fields whose domains have holes that are still conservative: for instance, $x/(x^{2}+y^{2})i+y/(x^{2}+y^{2})j$ is an example. Sławomir Biały (talk) 17:01, 12 September 2011 (UTC)

Time series in Excel

Hi, not sure if this is the right refdesk because it should be such an easy question. I have time series data, but not for every year, e.g. I have 1960, 1970, 1980, 1990, 2000 then every year to 2009. How, using Excel, can I predict the future trend? I can't even get Excel to plot the years at correct intervals on the x axis, i.e. to recognise that they are years rather than arbitrary labels. I do need to use Excel rather than any maths or stats software. I don't know if there is any significant trend in the data, or if there is, if it is linear or exponential. Thanks. Itsmejudith (talk) 09:00, 12 September 2011 (UTC)

Ducking the main question, I've just (to my surprise) got a fairly contemporary version of excel to do a nice enough graph of a data series such as you describe, by setting the date column to format=date - albeit I then had to enter 1/1/1960, etc ... it wasn't interested in a custom format of yyyy. It recognised it was dealing with dates, and the graph had an appropriate slope for the values I'd entered. Beyond that, I offer [1] fwiw. --Tagishsimon (talk) 09:20, 12 September 2011 (UTC)

There are two other ways to get Excel to correctly plot your time series against years. One method is to calculate interpolated values for the missing years - you could use simple linear interopolation or some form of polynomial interpolation if you wanted to avoid "corners" at 1970, 1980 etc. The other method is to tell Excel to create a scatter plot (older versions of Excel call it an "X-Y plot"). This tells Excel that your x-axis values are numeric as well as your y-axis values, so Excel will plot your data points correctly against numeric x and y axes; Excel can also connect these data points with smooth curves. Second method is simpler, but first method puts the interpolation method under your control. Predicting future values is extrapolation - I'm not sure to what extent this can be automated in Excel. I think you will probably need to select and define your extrapolation method first and then implement it as an Excel formula to calculate and then plot forecast values for future years. Excel does have built-in functiions to calculate the slope and intercept of a least-squares linear regression line. Gandalf61 (talk) 09:39, 12 September 2011 (UTC)

The Excel function for extrapolation is FORECAST(). - David Biddulph (talk) 15:01, 12 September 2011 (UTC)

On an Excel X-Y chart you can also add a Trendline (linear or other options). - David Biddulph (talk) 15:05, 12 September 2011 (UTC)

I don't know if ... any ... trend in the data ... is linear or exponential. If a nonconstant linear trend exists then sooner or later the variable turns negative. If all your data are positive you may want to take the logarithm before doing any extrapolation. Use your 1960, 1970, 1980, 1990, 2000 data to extrapolate into every year to 2009, to check consistency with your remaining data. Bo Jacoby (talk) 14:51, 12 September 2011 (UTC).

Many thanks, all. The main thing I was doing wrong, it seems, apart from not formatting the dates as Excel likes, was to try and use a line graph rather than a scatterplot. Obviously a scatterplot is more logical. I will then try FORECAST and the Trendline. Previously, I was getting completely different results from FORECAST and Trendline; if that happens again I may be back. When I have the spreadsheet worked out with my current data, I want to share it with others who will have their own data sets, so will have to think hard about whether to build in the possibility of taking logs, but thanks for that too. Itsmejudith (talk) 19:27, 12 September 2011 (UTC)

The FORECAST function does linear extrapolation along the least squares linear regression line of the set of x and y values that you prime it with. This should be identical with the linear option in Trendline, but Trendline offers you several other extrapolation methods as well. Gandalf61 (talk) 08:16, 13 September 2011 (UTC)