Calculus Of Several Variables

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Page By the same reason Euler, Leonhard Principia motus fluidorum 31 Aug Novi Comm. This system of notation was first used by the illustrious Fontaine ; as it appreciably simplifies calculation, I will also adopt it here. Kline, Morris Mathematical thought from ancient to modern times. Oxford University Press, Greenberg, John L.

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Alexis Fontaine's "fluxio-differential method'' and the origins of the calculus of several variables. In the years Fontaine produced an exposition of the calculus of several variables which we recognize to have been remarkably general for those years. Alexis Fontaine's integration of ordinary differential equations and the origins of the calculus of several variables. Fontaine was among those who pioneered the use of As a means of defining partial differential coefficients, the total differential was the earliest development in a recognizable calculus of several variables.

The Origins of Partial Differentiation. By , Euler knew that Fontaine had first introduced the notation. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Asked 5 years, 7 months ago.

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Table of Contents

II Differential Calculus of Several Variables. 14 Introduction to Differential Calculus. 15 Derivatives of Functions from R to Rn. Multivariable calculus is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions.

Delio Mugnolo Delio Mugnolo 2, 13 13 silver badges 36 36 bronze badges. Carlo Beenakker Carlo Beenakker The integrability condition existence of integrating factor goes back earlier than that: according to Euler's Opera Omnia, vol. Euler uses it in [, p.

In [ ], d'Alembert formally declared that this theorem does not belong to him but to Fontaine. I have corrected the source it was another one of Euler's publications and added a link to a discussion of Euler versus Clairaut. I found your ref. Viktor Blasjo Viktor Blasjo 1 1 bronze badge.

When did then functions of - say - two spacial variables begin to be studied? This means, in a sense, the function u t "speeds up" the curve, but keeps the curve's shape. This describes an infinite plane in R 3.

The concept of the level set or contour is an important one. Each of these level sets is a surface. Level sets in two dimensions may be familiar from maps, or weather charts. Each line represents a level set. For example, on a map, each contour represents all the points where the height is the same. On a weather chart, the contours represent all the points where the air pressure is the same.

Before we can look at derivatives of multivariate functions, we need to look at how limits work with functions of several variables first, just like in the single variable case. This means that by making the difference between x and a smaller, we can make the difference between f x and b as small as we want. Since this is an almost identical formulation of limits in the single variable case, many of the limit rules in the one variable case are the same as in the multivariate case. For f and g , mapping R m to R n , and h x a scalar function mapping R m to R , with.

Again, we can use a similar definition to the one variable case to formulate a definition of continuity for multiple variables.

Advanced Calculus of Several Variables

Finally, if f is continuous at p , g is continuous at f p , g f x is continuous at p. It is important to note that we can approach a point in more than one direction , and thus, the direction that we approach that point counts in our evaluation of the limit. It may be the case that a limit may exist moving in one direction, but not in another. We can't divide by vectors, so this definition can't be immediately extended to the multiple variable case. Nonetheless, we don't have to: the thing we took interest in was the quotient of two small distances magnitudes , not their other properties like sign.

It's worth noting that 'other' property of vector neglected is its direction. Now we can divide by the absolute value of a vector, so lets rewrite this definition in terms of absolute values. If we switch all the variables over to vectors and replace the constant which performs a linear map in one dimension with a matrix which denotes also a linear map , we have. A point on terminology - in referring to the action of taking the derivative giving the linear map A , we write D p f , but in referring to the matrix A itself, it is known as the Jacobian matrix and is also written J p f.

More on the Jacobian later. The consequence of this is that if f is differentiable at p , all the partial derivatives of f exist at p. However, it is possible that all the partial derivatives of a function exist at some point yet that function is not differentiable there, so it is very important not to mix derivative linear map with the Jacobian matrix especially in situations akin to the one cited.

Furthermore, if all the partial derivatives exist, and are continuous in some neighbourhood of a point p , then f is differentiable at p.

This has the consequence that for a function f which has its component functions built from continuous functions such as rational functions, differentiable functions or otherwise , f is differentiable everywhere f is defined. We use the terminology continuously differentiable for a function differentiable at p which has all its partial derivatives existing and are continuous in some neighbourhood at p. The chain rule for functions of several variables is as follows. Again, we have matrix multiplication, so one must preserve this exact order.

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Compositions in one order may be defined, but not necessarily in the other way. For simplicity, we will often use various standard abbreviations, so we can write most of the formulae on one line. This can make it easier to see the important details. Mostly, to make the formulae even more compact, we will put the subscript on the function itself. If we are using subscripts to label the axes, x 1 , x 2 …, then, rather than having two layers of subscripts, we will use the number as the subscript. If we are using subscripts for both the components of a vector and for partial derivatives we will separate them with a comma.

The most widely used notation is h x. Normally, a partial derivative of a function with respect to one of its variables, say, x j , takes the derivative of that "slice" of that function parallel to the x j 'th axis. More precisely, we can think of cutting a function f x 1 , From the definition, we have the partial derivative at a point p of the function along this slice as.

Instead of the basis vector, which corresponds to taking the derivative along that axis, we can pick a vector in any direction which we usually take as being a unit vector , and we take the directional derivative of a function as. The partial derivatives of a scalar tell us how much it changes if we move along one of the axes. What if we move in a different direction? This is the dot product of dr with a vector whose components are the partial derivatives of f , called the gradient of f. We can form directional derivatives at a point p , in the direction d then by taking the dot product of the gradient with d.

Notice that grad f looks like a vector multiplied by a scalar. This particular combination of partial derivatives is commonplace, so we abbreviate it to. We can write the action of taking the gradient vector by writing this as an operator. The level sets of h are concentric circles, centred on the origin, and. If dr points along the contours of f , where the function is constant, then df will be zero.

Since df is a dot product, that means that the two vectors, df and grad f , must be at right angles, i. For any pair of constants, a and b , and any pair of scalar functions, f and g. Since it's a vector, we can try taking its dot and cross product with other vectors, and with itself.

This dot product is called the divergence. Div u tells us how much u is converging or diverging. It is positive when the vector is diverging from some point, and negative when the vector is converging on that point.

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Later in this chapter we will see how the divergence of a vector function can be integrated to tell us more about the behaviour of that function. If we reverse the order we get. This cross product is called the curl. Curl u tells us if the vector u is rotating round a point. The direction of curl u is the axis of rotation.

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We can then extend the definition of curl u to two-dimensional vectors. This two dimensional curl is a scalar.

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In four, or more, dimensions there is no vector equivalent to the curl. These vectors are tangent to circles centred on the origin, so appear to be rotating around it anticlockwise. Later in this chapter we will see how the curl of a vector function can be integrated to tell us more about the behaviour of that function.