what is vector calculus?
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Vector calculus is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. It consists of a suite of formulas and problem solving techniques very useful for engineering and physics.\n\nWe consider vector fields, which associate a vector to every point in space, and scalar fields, which associate a scalar to every point in space. For example, the temperature of a swimming pool is a scalar field: to each point we associate a scalar value of temperature. The water flow in the same pool is a vector field: to each point we associate a velocity vector.
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The notion of a Vector space, a mathematical system of a certain kind, has developed into one of the richest concepts in both pure and applied mathematics.\n\nThe concept of vector space pervades mathematics, engineering and the sciences. The questioner is surely familiar with vectors in the form of directed line segment or columns of numbers studied in calculus and mechanics. \n\nThe scope of a general notion of vector is much broader than these examples. For example, functions, matrices and the operation of differentiation will turn out to be vectors.\n\nVector spaces turn up in a wide variety of studies. For example, vector space arise naturally in the study of solutions of systems of equations, geometry in 3-space, solutions of differential equations, finite elements, discrete and continuous Fourier transforms, quantum mechanics, approximation theory and the study of a wide variety of other mathematical and scientific disciplines
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It is linear on intervals, on (- infinite, 0) and on (0, +infinite) but not on the whole R. \n\nTry plotting it, you get a reversed V, not a straight line.
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Let's start with an example of a vector space. An example is a plane. If we take any two points (we can make them vectors by drawing the points through the origin) in the plane and add them, the result will still be in the plane. If we take any scalar multiple of a vector in the plane, the result will still be in the plane. In fact, the resulting vector will be parallel to the original vector. \n\nIt's these two properties that make a vector space a vector space. In fact, what I said about the plane holds for any n-dimensional (n is any whole number that can be imagined) space. Further generalizations can be made to other examples that satisfy the addition and scalar multiplication properties mentioned above. But staying in the same space when adding or multiplying by a scalar are the key properties of a vector space.
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You need to apply vectors.
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Newtonian Mechanics.\n\nI don't know why people are giving all the credits to Newton for Calculus, when in fact he and Liebniz merely collected all the works of previous mathematicians, combined it with their own works, and compiled it independently.
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Try here:/\n\nhttp://archives.math.utk.edu/visual.calculus/
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good luck man... http://mathforum.org/library/drmath/sets/high_calculus.html
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Calculus on Manifolds \nMichael Spivak's Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus (1965) is a text treating analysis in several variables in Euclidean spaces and on differentiable manifolds. Notable features include a long problem entitled "A First Course in Complex Analysis" and the book's cover, a reproduction of Lord Kelvin's letter to Stokes in which Stokes's theorem is first stated.
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U•V = |U||V|cosα  Where |U| is the norm of U and α is the angle between the vectors.\n\nIt also boils down to:\nU•V = x1y1+x2y2\nU•V= (-4)(1) + (6)(5)\nU•V = 26
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In cartography, as in all mappings, a vector implies direction & length (or force). I'm not familiar with other uses of the word... I usually have used the term in connection "cross-impact matricies." Hope that was helpful.
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Geez, this is like asking, "How do you do math?"  Calculus is an entire field within the discipline of mathematics.  There is so much involved, you can summarize the goal of calculus but not the "how."\n\nThere are basically two branches of calculus:  differential calculus and integral calculus.  In differential calculus, basically all you are doing is finding the slope of a curve.  With integral calculus, you are basically finding the area under a curve.  (Sound simple, huh?)  Both of these have MANY applications, so there is a lot of "how" that goes on in calculus courses.\n\nAt the university I graduated from, there are five semesters of calculus, not counting the "side" courses like differential equations.  Remember that two years of high-school algebra is compacted into one semester of college algebra, and you can get an idea of the scope of calculus.\n\nIf you do have specific questions about specific types of calculus problems, we can be more helpful with the "how."  You have just given us a question that is WAY too vague to address well.
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1. For two given points P(p1, p2, p3) and Q(q1, q2, q3), the vector v between them is v = (x, y, z) = (q1 - p1, q2 - p2, q3 - p3).\n\nThe magnitude or length of v, denoted by ||v|| is\n||v|| = sqrt[ x^2 + y^2 + z^2 ]\n\nJust do that for every line that you have.\n\n2. This one's a little harder. Let's define the first line as the vector u, and your second line that you're using as the vector v.\n\nYou first need to write each one as a unit vector, which means dividing all of the values through by a value so that the magnitude will be 1. For each of these vectors, this means dividing by the length of the vector.\n\nAs an example, if I wanted to turn w = (1, 1, 1) into a unit vector, I would notice that the magnitude is ||w|| = sqrt[3] and divide all the values through by that to get (1 / sqrt[3], 1 / sqrt[3], 1 / sqrt[3]).\n\nSo, let us say that the new unit vectors are u / ||u|| = (a, b, c) and v / ||v|| = (x, y, z)\n\nWhat these values give you now are called the direction cosines of each vector. If A is the angle between u and the x axis, then a = cos A, and similarly for b with the y-axis and c with the z-axis.\n\nSo, to find the angles between each vector and the x, y, and z axes (also called the direction angles), you take the inverse cosine (cos -1) of the components (values) of the vector.\n\nLet us say that after taking the inverse cosine of u / ||u|| = (a, b, c) = (cos A, cos B, cos C), you get (A, B, C), and the inverse cosine of v / ||v|| gives you (X, Y, Z).\n\nThe angles between the two vectors in the x, y, and z directions are (X - A, Y - B, Z - C).\n\nSo the main steps are finding the magnitude of each vector, normalizing each vector (making them into unit vectors), taking the inverse cosine of each vector, and subtracting the two vectors.\n\nHope that helps.
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Integration is one of the two major operations of calculus (the other being the inverse, differentiation. There are indefinite and definite types of integration. When the integral of a function is expressed symbolically (as another function), this is the indefinite integral (e.g., the ind. int. of y=2x is y=x^2 + a constant). When the integral is numerically evaluated over some range of x, this is the definite integral. Much of time-domain simulation consists of successive evaluation and accumulation of the definite integral of some function, or "time stepping". Typically, each time step starts with evaluation of the 2nd derivative of something, let's say position (whose 2nd deriv. is acceleration), accumulation of the 1st deriv. (velocity) as the previous value + time-step size * 2nd deriv., then accumulation of the 0th deriv. (position) as the previous value + time-step size * 1st deriv. Of course it's a bit more complex than that, and many integration formulas exist to minimize errors.\nAn "integration" function could be something like what I've described for time-stepping, or it could be a much more sophsticated function that accepts a function definition and gives you the indefinite integral in symbolic form. So you need more information about what kind of integration function is desired.
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Curl, divergence, gradient.
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Whenever the speed and direction of motion are involved, we experience vectors, and our minds unconsciously solve vector equations that allow us to intercept a moving object like a thrown ball. This enables us to run toward where the ball will be when it comes down, rather than chasing toward it throughout its flight. Not only do we predict the ball's path based on its velocity and gravitational acceleration, but we compensate for wind disturbance as well by incorporating information based on its drift rate in our visual frame. It's true we don't deal with it as a vector math situation, with numerical values in 3 coordinates. Instead we can be said to view the situation as a 2-coordinate problem (height and ground track) with lateral adjustments, and we solve it continuously and holistically.
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questions are very easy. You must try them by yourself. I will only give hints-\n1> Instantaneous accl  = dv/dt\nAvg accl = (v2-v1)/(t2-t1)\n\n2> draw the vectors, take the components separately along x and y axes.\nremember accl component along x axis increases velocity along x axis, and accl component along y axis increases velocity along y axis.\n3> use the basic kinematics equations like v^2=u^2+2as etc