plane_vector_fld

# plane_vector_fld - MIT OpenCourseWare http:/ocw.mit.edu...

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V. VECTOR INTEGRAL CALCULUS V1. Plane Vector Fields 1. Vector fields in the plane; gradient fields. We consider a function of the type where M and N are both functions of two variables. To each pair of values (xo, yo) for which both M and N are defined, such a function assigns a vector F(xo, yo) in the plane. F is therefore called a vector function of two variables. The set of points (x, y) for which F is defined is called the domain of F. To visualize the function F(x, y), at each point (xo, yo) in the domain we place the corresponding vector F(xo, yo) so that its tail is at (xo, yo). Thus each point of the domain is the tail end of a vector, and what we get is called a vector field. This vector field gives a picture of the vector function F(x, Y). Conversely, given a vector field in a region of the xy-plane, it determines a vector function of the type (I), by expressing each vector of the field in terms of its i and j components. Thus there is no real distinction between "vector function" and "vector field". Mindful of the applications to physics, in these notes we will mostly use "vector field". We will use the same symbol F to denote both the field and the function, saying "the vector field F" , rather than "the vector field corresponding to the vector function F". We say the vector field F is continuous in some region of the plane if both M(x, y) and N(x, y) are continuous functions in that region. The intuitive picture of a continuous vector field is that the vectors associated to points sufficiently near (xo, yo) should have direction and magnitude very close to that of F(xo, yo) - in other words, as you move around the field, the vectors should change direction and magnitude smoothly, without sudden jumps in size or direction. In the same way, we say F is differentiable in some region if M and N are differentiable, that is, if all the partial derivatives exist in the region. We say F is continuously differentiable in the region if all these partial derivatives are themselves continuous there. In general, all the commonly used vector fields are continuously differentiable, except perhaps at isolated points, or along certain curves. But as you will see, these points or curves affect the properties of the field in very important ways.
2 V. VECTOR INTEGRAL CALCULUS Where do vector fields arise in science and engineering?

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## This note was uploaded on 05/04/2011 for the course MATH 18.02 taught by Professor Auroux during the Spring '08 term at MIT.

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