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MJC684
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Questions on a "modern" theory of the differential from a gem of an old Calc text
I discovered a gem of an multivariable calculus text from the 60s called "Modern Multidimensional Calculus" by M. E. Munroe. First off is anyone familiar with this text by any chance? It presents variables and differentials in way that I have never seen before and was wondering if this how these concepts are defined in Differential Geometry perhaps.
The variables x and y are treated not simply as variables but as mappings. The mapping x and the mapping y map points in the plane to their abscissas and ordinates respectively. So x is x(p) and y is y(p). It then distinquishes these coordinate mappings from the type that is usually presented in elementary calculus that of f(x). A "function" in this text is a mapping that maps numbers into numbers as opposed to points, (physical/geometrical objects) to numbers. In this way f(x) is actually f[x(p)]. It claims that in the modern theory of the differential it is absolutely essential to recognize x and y as mappings themselves.
The text says things that I am not 100% sure that I'm understanding completely like " Should the equation y = F[x(p)] be taken by itself and regarded as an assertion that y and F[x(p)] are two symbols for the same mapping? Unfortunately the answer is no in analytic geometry and yes in calculus"
Why is that?
The way the differential of x and y is defined is just as interesting and new to me as well. It makes a lot more sense than all the double talk concerning differentials and increments in the usual calculus texts. I won't get into the definition it gives for the differential just yet unless someone who is qualified takes an interest in this posting.
My question is has anyone seen this type treatment of calculus before? Is this perhaps the way the x and y variables are treated in more advanced mathematics like Diff Geometry or just this authors own special treatment of the subject? worth studying despite the ill be covering the same material next semester in the traditional way?
If anyone is interested ill get into the differential definition. Thanks in advance
I discovered a gem of an multivariable calculus text from the 60s called "Modern Multidimensional Calculus" by M. E. Munroe. First off is anyone familiar with this text by any chance? It presents variables and differentials in way that I have never seen before and was wondering if this how these concepts are defined in Differential Geometry perhaps.
The variables x and y are treated not simply as variables but as mappings. The mapping x and the mapping y map points in the plane to their abscissas and ordinates respectively. So x is x(p) and y is y(p). It then distinquishes these coordinate mappings from the type that is usually presented in elementary calculus that of f(x). A "function" in this text is a mapping that maps numbers into numbers as opposed to points, (physical/geometrical objects) to numbers. In this way f(x) is actually f[x(p)]. It claims that in the modern theory of the differential it is absolutely essential to recognize x and y as mappings themselves.
The text says things that I am not 100% sure that I'm understanding completely like " Should the equation y = F[x(p)] be taken by itself and regarded as an assertion that y and F[x(p)] are two symbols for the same mapping? Unfortunately the answer is no in analytic geometry and yes in calculus"
Why is that?
The way the differential of x and y is defined is just as interesting and new to me as well. It makes a lot more sense than all the double talk concerning differentials and increments in the usual calculus texts. I won't get into the definition it gives for the differential just yet unless someone who is qualified takes an interest in this posting.
My question is has anyone seen this type treatment of calculus before? Is this perhaps the way the x and y variables are treated in more advanced mathematics like Diff Geometry or just this authors own special treatment of the subject? worth studying despite the ill be covering the same material next semester in the traditional way?
If anyone is interested ill get into the differential definition. Thanks in advance