Math 251 Final Review Pack

6 cylinders and quadratic surfaces given some

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Unformatted text preview: iven some constant, k Planes: Trace Curves: (Parabola) (Circle) Cylinder: A surface obtained by parallel lines that go through the points of a given cure Going back to 2D: (Ellipse) (Hyperbola) Quadratic Surface: Any surface whose equation in 3D is given by a quadratic surface Elliptic Parabolid: A shape that contains two parabolas and one ellipse as its trace curves Hyperbolic Parabolid: A shape that contains two parabolas and a hyperbola as its trace curves Hyperbolid of One Sheet: A shape that contains two hyperbolas and one ellipse as its trace curves Hyperbolid of Two Sheets: A hyperbolid of one sheet which is multiplied by -1 Cone: A shape that contains two sets of lines and a ellipse as its trace curves, its equation always equals to 0 8 Chapter 13: Vector Functions 9 13.1 Vector Functions and Space Curves A vector function is a function that takes in one input and produces multiple outputs Definition (Limit): The limit of a vector function is defined by: Definition (Continuity): A vector function r is continuous at t=a if: Definition (Space Curves): The set of all points (x,y,z) such that: For some functions f,g,h The distance from a plane to some point 10 is: 13.2 Derivatives and Integrals of Vector Functions Definition (Derivative): The derivative of r(t) is defined by: If the limit exists then r(t) is differentiable Tangent Vector: Tangent Line: the line that goes through the point r(t) and has the direction of r’(t) Unit Tangent Vector: Theorem: If , then Properties: If u and v are differentiable functions and c is some scalar and f is a real valued function then: 1) 2) 3) 4) 5) 6) Definition (Definite Integral): The definite integral of r(t) is defined by: Where 11 13.3 Arc Length and Curvature Recall: Length of a plane curve parameterized by Length: This formula extends to is given by: : Consider or equivalently We may rewrite the previous formula as: A curve may have different parameterizations Length is independent of the choice of parameterization Given: : Arc length function: It is often useful to parameterize a curve with respect to its arc length Smooth parameterization: A parameterization continuous and on is called smooth on an interval if is Smooth curve: A curve that has a smooth parameterization. A smooth curve has no sharp points. For a smooth curve: Recall: is the unit tangent vector 12 Curvature measures the rate of change of the unit tangent vector when the curve is parameterized using the arc length function. Curvature is defined as: Suppose the curve is parameterized with : Using the chain rule: Note: We get: Theorem: Normal and Binormal Vectors Consider a curve Since and , the unit tangent vector then Normal plane at point Osculating plane at point is the plane through with normal vector is the plane through 13 with normal vector 13.4 Motion in Space Suppose describes the motion of some particle in space: The displacement distance can be defined as: For Claim: , lies in the osculating plane We can rewrite as a linear combination of T and N: Recall: We can rearrange this equation: This also proves that the acceleration vector lies in the osculating plane. Kepler’s Laws Astronomers Tycho Brache and Johannes Kepler formulated the laws of planet...
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