Negative one, two, three, four, five, six, Now first imagine that this was if we just drew this vector where we count starting from the origin, Nine times one is nine so one minus nine is negative eight. Two cubed is eight nine times two is 18 so eight minus 18 is negative 10 negative 10 and then one cubed is one, So lets walk through anĮxample of what I mean by that so if we actually evaluate F at one,two X is equal to one Y is equal to two so we plug in two cubed whoops, two cubed minus nine times two up here in the X component and then one cubed minus nine times Y nine times one, excuse me down in the Y component. Vector that it outputs and attach that vector to the point. This here's our X axis this here's our Y axis and for each individual input point like lets say one,two so lets say we go to one,two I'm gonna consider the So I'll draw these coordinate axes and just mark it up, Visualize a function like this with a graph it would be really hard because you have twoĭimensions in the input two dimensions in the output so you'd have to somehow visualize this thing in four dimensions. Looking kind of similar they don't have to be, I'm The Y component of the output will be X cubed minus nine X. Y cubed minus nine Y and then the second component, So here I'm gonna write a function that's got a two dimensional input X and Y, and then its output is going to be a two dimensional vector and each of the components That have the same number of dimensions in their Is its pretty much a way of visualizing functions It comes up with fluidįlow, with electrodynamics, you see them all over the place. That come up all the time in multi variable calculus,Īnd that's probably because they come up all the time in physics. Use double, triple and line integrals in applications, including Green's Theorem, Stokes' Theorem, Divergence Theorem and Fundamental theorem of line integrals.Everyone, so in this video I'm gonna introduce vector fields. Synthesize the key concepts differential, integral and multivariate calculus.Įvaluate double integrals in Cartesian and polar coordinates evaluate triple integrals in rectangular, cylindrical, and spherical coordinates and calculate areas and volumes using multiple integrals. Graphically and analytically synthesize and apply multivariable and vector-valued functions and their derivatives, using correct notation and mathematical precision. Perform vector operations, determine equations of lines and planes, parametrize 2D & 3D curves. Upon completion of this course, students should be able to: Recognize and apply Fundamental theorem of line integrals, Green’s theorem, Divergence Theorem, and Stokes’ theorem correctly. Set up and evaluate double and triple integrals using a variety of coordinate systems Įvaluate integrals through scalar or vector fields and explain some physical interpretation of these integrals Recognize and apply the algebraic and geometric properties of vectors and vector functions in two and three dimensions Ĭompute dot products and cross products and interpret their geometric meaning Ĭompute partial derivatives of functions of several variables and explain their meaning Ĭompute directional derivatives and gradients of scalar functions and explain their meaning Ĭompute and classify the critical points Recognize and sketch surfaces in three-dimensional space
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