9. On the last day of class, we described how two curves in R? being tangent to each other corresponded to their normal vectors being parallel (and this helped explain why Lagrange multipliers work). Now we try to generalize this idea of tangency in various ways. (a) exactly one point P in R³. Do NOT put one sphere inside the other one. Draw a picture of two spheres of different sizes being tangent to each other at Suppose we have two surfaces S1 and S2 which intersect at some point P : (a1,b1, c1) (for S1) and (b) (xo, Y0, z0), and further suppose that the normal vectors nı = n2 = (a2, b2, c2) (for S2) at P are parallel (and non-zero). Prove that the tangent planes TpS1 and TPS2 are the same by transforming the scalar equation for TpSı into the scalar equation for TPS2. Hint: This should be a one-step algebraic transformation based upon using the algebraic definition of parallel vectors.
Arc Length
Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length.
Line Integral
A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.
Triple Integral
Examples:
college level
topic: R^3, third dimension, spheres, tangency, drawing models in vector calculus

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