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Multivariate Calculus Study Guide

Introduction to Vectors

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Multivariable calculus extends the notion of differentiation and integration in three dimensional space. New concepts such as partial derivatives, directional derivatives, line integration and multiple integrals are introduced.

When plotting in 3 dimensional space, there needs to be new notation for various concepts that will be introduced. Multivariable calculus also introduces vectors, which may have been introduced in physics. Calculus dealing with vectors will take into consideration not only magnitude but direction as well.

Multivariable calculus is a broad field that applies to physics and other sciences more so than single variable calculus. Many of the theorems provided in vector calculus are essential for solving problems in physics, which are mostly multidimensional. Multivariable calculus studies the rates of change and works in more than two dimensions.

In single variable calculus, an object might move through the air and we can calculate its velocity at any given point. Vector calculus was developed from quaternion analysis by J. Willard Gibbs and Oliver Heaviside near the end of the 19th century, and most of the notation and terminology was established by Gibbs and Edwin Bidwell Wilson in their book, Vector Analysis.

In the conventional form using cross products , vector calculus does not generalize to higher dimensions, while the alternative approach of geometric algebra , which uses exterior products does generalize, as discussed below.

A scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number or a physical quantity. Examples of scalar fields in applications include the temperature distribution throughout space, the pressure distribution in a fluid, and spin-zero quantum fields, such as the Higgs field. These fields are the subject of scalar field theory. A vector field is an assignment of a vector to each point in a subset of space.

Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force , such as the magnetic or gravitational force, as it changes from point to point. In more advanced treatments, one further distinguishes pseudovector fields and pseudoscalar fields, which are identical to vector fields and scalar fields except that they change sign under an orientation-reversing map: This distinction is clarified and elaborated in geometric algebra, as described below.

The algebraic non-differential operations in vector calculus are referred to as vector algebra , being defined for a vector space and then globally applied to a vector field. The basic algebraic operations consist of:. Also commonly used are the two triple products:.

The three basic vector operators are:. A quantity called the Jacobian matrix is useful for studying functions when both the domain and range of the function are multivariable, such as a change of variables during integration. The three basic vector operators have corresponding theorems which generalize the fundamental theorem of calculus to higher dimensions:.

Linear approximations are used to replace complicated functions with linear functions that are almost the same. For a continuously differentiable function of several real variables , a point P that is a set of values for the input variables, which is viewed as a point in R n is critical if all of the partial derivatives of the function are zero at P , or, equivalently, if its gradient is zero.

The critical values are the values of the function at the critical points. If the function is smooth , or, at least twice continuously differentiable, a critical point may be either a local maximum , a local minimum or a saddle point.

The different cases may be distinguished by considering the eigenvalues of the Hessian matrix of second derivatives.

By Fermat's theorem , all local maxima and minima of a differentiable function occur at critical points. Therefore, to find the local maxima and minima, it suffices, theoretically, to compute the zeros of the gradient and the eigenvalues of the Hessian matrix at these zeros.

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Vector Calculus Quite often math tends to be one of the most difficult areas to study, and this is especially true when it comes to tackling vector calculus homework/5(94).

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Feb 12,  · Homework Help: Vector calculus Feb 12, #1. danny_manny. 1. The problem statement, all variables and given/known data Use the given information to find the position and velocity The constant, c, you have after integrating a(t) is a vector constant, c. SammyS, Feb 12, Our mathematics tutors can help with all your projects, large or small, and we challenge you to find better online calculus tutoring anywhere. Get College Homework Help. I Need Written Solutions.