Vector (good one)

Vector (good one) - Vector Calculus Study Guide Charlie...

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Unformatted text preview: Vector Calculus Study Guide Charlie Egedy April 7, 2009 1 Vectors in R3 Note that to get definitions and theorems for R2 , simply make the third coordinate zero, since this will restrict attention to the x-y coordinate plane in R3 . • If (a, b, c) is a point in R3 , then ￿a, b, c￿ is a position vector, that is, one that begins at the origin and ends at the point, with orientation toward the point. • Given two points, we can define a vector whose coordinates are computed as the differences between corresponding coordinates. Orientation is away from the point whose coordinates were subtracted. • We define vector addition by adding corresponding coordinate, and scalar multiplication by multiplying each coordinate by the same scalar. Thus, ￿a, b, c￿ + r ￿c, e, f ￿ = ￿a + rd, b + re, c + rf ￿. • Vector addition has the geometric interpretation of constructing the diagonal of a parallelogram defined by the two vectors, oriented outward from the common point of origin. The difference between two vectors is the other diagonal, with orientation so that the convention on vector addition is maintained. • Vector addition is commutative and associative, there is a zero vector, and each vector has an additive inverse. Additionally, scalar multiplication is distributive over vector addition. ￿ • If ￿ = ￿a1 , a2 , a3 ￿, then ￿￿ ￿ = a2 + a2 + a2 . a a 1 2 3 • Given any vector other than the zero vector, we can define a unit vector having the same orienv tation as the given vector. Thus, given ￿ , the corresponding unit vector is e￿ = ￿￿ ￿ . v v ￿ v ˆ • Sometimes it will be useful to use the notation ˆ = ￿1, 0, 0￿, ˆ = ￿0, 1, 0￿ and k = ￿0, 0, 1￿. i j • Triangle inequality always applies: ￿￿ + w￿ ≤ ￿￿ ￿ + ￿w￿. v￿ v ￿ • We orient R3 according to the right hand rule. • An equation for a line can be written as a vector valued function of position vectors that are oriented toward points on the line by writing ￿(t) = ￿x0 , y0 , z0 ￿ + t ￿v1 , v2 , v3 ￿. The first vector is a r position vector that hits some point on the line and the second vector gives the line its orientation. • The parametric form of the equation for a line comes from writing the coordinates explicitly. Thus x = x0 + tv1 , y = y0 + tv2 and z = z0 + tv3 . y x • The symmetric form for a line comes by equating t in the above equations. Thus, x−1 0 = y−2 0 = v v z −z0 v3 . If any of the coordinates for the orienting vector is zero, this means that the cooresponding coordinate does not change with t, so we leave that variable out of the string of equalities, writing the constant that that coordinate is equal to apart from the equation. ￿ ￿ • Given points P and Q, we can parameterize a line by writing ￿(t) = (1 − t)OP + tOQ. The point r 1 midway between P and Q then occurs when t = 2 . 1 • We can write the equation of a sphere having radius R, centered at (a, b, c) by (x − a)2 + (y − b)2 + (z − c)2 = R2 . • We define dot product between two vectors as the sum of corresponding products. Thus, ￿ · w = v￿ v1 w1 + v2 w2 + v3 w3 . • The dot product of a vector with itself is the square of its magnitude. v￿ • The angle between vectors is given by cos θ = ￿￿￿ ·w￿ ￿ . If the numerator is positive, the angle is v ￿￿w acute; if negative the angle is obtuse. If zero, the angle is right. In the last case we say that vectors are perpendicular or orthogonal. • We can resolve one vector in terms of another by first computing the magnitude of the parallel resolution and then multiplying this by a unit vector in the desired direction. Thus compw (￿ ) = ￿v ￿ ·w v￿ ￿ ·w v￿ ￿v ￿ ￿w￿ is the component, and the projection itself is projw (￿ ) = ￿w￿ ew , as the parallel component of ￿ ￿ ￿ in the direction of w. We compute the perpendicular component of the resolution by subtracting v ￿ the parallel component from the original vector being resolved. • The cross product of two vectors is a vector that is perpendicular to the two ￿ ￿ˆ ￿i define the cross product as the determinant of a matrix. Thus, ￿ × ￿ = ￿ u1 uv￿ ￿ v1 given vectors. We ￿ ˆ ˆ k￿ j ￿ u2 u 3 ￿. ￿ v2 v3 ￿ • The magnitude of the cross product is the product of the magnitudes of the two vectors times the sine of the angle between them. The orientation of the cross product is given by the right hand rule. • Cross product is anticommutative (that is, swapping the order induces a minus sign). The cross product of a vector with itself is the zero vector, and cross product distributes over addition. • The area of a parallelogram is the magnitude of the cross product of the two vectors that define it. The volume of a parallelpiped is the triple product of the three vectors that defines it. • The equation ax + by + cz = d is a plane with normal ￿a, b, c￿ containing the point (x0 , y0 , z0 ), where ax0 + by0 + cz0 = d. • Angles between planes are defined to be the angles between their normals. • Parallel planes have parallel normals. • To find the point of intersection between a line and a plane, plug the parametric equations of the line into the line into the equation of the plane and solve for t. • To find the point of intersection of two lines, give them different parameter names and then equate the corresponding coordinates, solving for the two parameters. If no solution exists, then the lines are parallel (with parallel orienting vectors) or they are skew. • The distance between two planes is the projection of a vector connecting points in each plane with a common normal for the two planes. • The distance between skew lines is the projection of a vector connecting points in each line with a normal defined to be the cross product of the vectors orienting the two lines. • To find the equation of a line of intersection between two planes, first compute the orienting vector as the cross product of the normals for the two planes. Then find a point common to the two planes by solving the equations of the plane simultaneously, choosing any point of convenience. • Given three points, the equation of the plane containing them can be found by first defining two vectors oriented in the plane as differences between point coordinates, computing the cross product of these two vectors to get the plane’s normal, then computing d using one of the three points. 2 • There are many variations on the theme contained in Rogawski’s problem set 12.5. Work them all. • Quadric surfaces for equations with no cross terms: – If two variables are missing, then the surface is either a plane or two planes. – If one variable is missing, then the surface is a cylinder: parabolic, elliptic, circular or hyperbolic, depending upon the conic section depicted by the equation if it were in R2 . – If no variables are missing: ∗ If none are squared, then the surface is plane. ∗ If one is squared, then the surface is a parabolic cylinder, with rulings that are not parallel to any coordinate axis. ∗ If two are squared, then the surface is either a paraboloid with either circular or elliptical cross sections, or it is a hyperbolic paraboloid. ∗ If three are squared and the constant on the RHS is positive, then the surface is: · An ellipsoid if all signs are positive. It is a sphere if all coefficients are equal, a prolate spheriod if two are equal, representing minor axes, or an oblate spheroid if two are equal representing major axes. · A one-sheet paraboloid if only one sign is negative. · A two sheet paraboloid if two signs are negative. ∗ If the RHS is zero, then it is a point if all three signs are positive, and a cone if one or two of the signs are negative. 2 Vector Calculus • A vector valued function is any vector that depends upon a parameter. • A line is a vector valued function, as is the helix ￿(t) + ￿cos(t), sin(t), t￿. r • We take limits or derivatives or integrals of vector valued functions by taking llimits or derivatives or integrals on a coordinate basis. • We compute speed of a vector valued function as the magnitude of the derivative of the vector valued function. The arc length is the integral of the speed. • In certain cases, it is convenient to reparameterize based upon arc length. This will happen when the arc length is an invertible function of the original parameter. • The unit tangent vector is the unit vector in the direction of the derivative of the vector valued function. • Curvature is the derivative of the unit tangent vector with respect to arc length. It can also be calculated by computing the magnitude of the derivative of the unit tangent vector and dividing by the speed function. The radius of curvature is the multiplicative inverse of the curvature. • We can also compute curvature by knowing both the first and second derivatives of the vector valued function. We divide the magnitude of the cross product of these two vectors by the cube of the magnitude of the derivative of the vector valued function. • The unit normal is the derivative of the unit tangent vector, normalized. • The binormal is the cross product of the unit tangent vector and the unit normal. • Given a vector valued fucntion: – The velocity vector is the derivative of the vector valued function. Its magnitude is speed. – The acceleration vector is the derivative of the velocity vector. – The acceleration vector can be resolved along the unit tangent and unit normal vectors. 3 3 Partial Derivatives • A partial derivative is a derivative of a function having more than one independent variable, taken by treating all but one of the variables as constants and then taking the derivative with respect to the nonconstant variable. • If derivatives are continuous, then mixed partials can be computed by taking derivatives in any order. 4 ...
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