Use double, triple and line integrals in applications, including Green's Theorem, Stokes' Theorem, Divergence Theorem and Fundamental theorem of line integrals. 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. Vector Calculus 4th Edition ISBN: 9780321780652 Susan J. 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 The goal is to apply one vector to another.Recognize and sketch surfaces in three-dimensional space You've seen the dot product equation everywhere:Īnd also the justification: "Well Billy, the Law of Cosines (you remember that, don't you?) says the following calculations are the same, so they are." Not good enough - it doesn't click! Beyond the computation, what does it mean? With the quaternions (4d complex numbers), the cross product performs the work of rotating one vector around another (another article in the works!).We will pick the three best parts of midterms and finals to calculate your grade. “Multiply” two vectors when only perpendicular cross-terms make a contribution (such as finding torque). Text:Marsden and Tromba, Vector calculus, 6th edition Calculation of Grade Your grade will depend on your scores in homeworks, midterms and final as follows: homework (28), 2 midterms (24 each) and the final, part A and B (24 for each part).Determine if two vectors are orthogonal (checking for a dot product of 0 is likely faster though).Find the signed area spanned by two vectors.Find the direction perpendicular to two given vectors. (Try it: using your right hand, you can see x cross y should point out of the screen). In a computer game, x goes horizontal, y goes vertical, and z goes “into the screen”. The Unity game engine is left-handed, OpenGL (and most math/physics tools) are right-handed. I never really memorized these rules, I have to think through the interactions. This completed grid is the outer product, which can be separated into the:ĭot product, the interactions between similar dimensions ( x*x, y*y, z*z)Ĭross product, the interactions between different dimensions ( x*y, y*z, z*x, etc.) Taking two vectors, we can write every combination of components in a grid:
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