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Multivariable Calculus – Questions/Answers
Distance between two points (a,b,c) and (x,y,z) ✔Ans - d = sqrt((x-a)^2 + (y- b)^2 + (z-c)^2) cross sections vs level curves ✔Ans - cross sections: vertical slices of the graph of f(x,y) formed using vertical planes x=c (f(c,y)=z) or y=c (f(x,c)=z). Level curves: horizontal slices of f(x,y) using horizontal planes z=c (f(x,y) =c). Both used to plot graphs of functions with multiple variables. contour diagrams ✔Ans - a family of graphs of the equation f(x,y) = c plotted in the xy-plane, for set values of c, usually labeled by the values. Values of c usually plotted in equal increments. (like topographical map) m and n ✔Ans - m= slope in positive x direction dz/dx (holding y constant) n = slope in positive y direction dz/dy (holding x constant) equation for plane passing through the point (x0, y0, z0) with slope m in +x direction and n in +y direction ✔Ans - z-z0 = m(x-x0) + n(y-y0) Point-slope: f(x,y) = z = z0 + m(x-x0) + n(y-y0) Slope-int: f(x,y) = z = c + mx + my. c = z0- m(x0)-n(y0) What does f(x,y) = c (constant) look like? ✔Ans - horizontal plane Level surfaces ✔Ans - Used for visualizing a function of three variables w = f(x,y,z). For various constants w = c, plot the surface whose graph is f(x,y,z) = c. Creates a 3D contour diagram elliptical paraboloid ✔Ans - z = x^2/a^2 + y^2/b^ hyperbolic paraboloid ✔Ans - z = -x^2/a^2 + y^2/b^ Ellipsoid ✔Ans - x^2/a^2 + y^2/b^2+ z^2/c^2 = 1 hyperboloid of one sheet ✔Ans - x^2/a^2 + y^2/b^2- z^2/c^2 = 1 hyperboloid of two sheets ✔Ans - x^2/a^2 + y^2/b^2- z^2/c^2 = -
cone ✔Ans - x^2/a^2 + y^2/b^2 - z^2/c^2 = 0 Plane ✔Ans - ax + by + cz = d cylindrical surface ✔Ans - x^2 + y^2 = a^ parabolic cylinder ✔Ans - y = ax^ limit ✔Ans - lim (x,y) -> (a,b) f(x,y) = L , if f(x,y) can be made as close to L as we please whenever the distance from point (x,y) to the point (a,b) is sufficiently small, but not zero Finding the limit of a function along a line, or another function (ex. y = mx, y = x^2) ✔Ans - Plug function into each y in the larger function and evaluate the limit at the point. If the limits approaching a point along multiple different functions differ the limit does not exist. displacement vector ✔Ans - vector representing displacement i.e. change from a point P to point Q. unit vector ✔Ans - has magnitude of 1 unit how to add, subtract, scalar multiply vectorys ✔Ans - add, subtract and scalar multiple by components magnitude ✔Ans - ||v|| = sqrt(v1^2 + v2^2 + v3^2) ||cv|| = |c|||v|| algebraic properties of vectors: commutativity, associativity, distributivity, identities ✔Ans - comm: v + w = w + v assoc:(u + v) + w = u + (v + w) a(bv) = (ab)v dist: (a + b)v = av + bv a(v + w) = av + aw ident:1v =v v + 0 = v 0v = o
fy(a,b) = lim k-->0 f(a,b + k) - f(a,b)/k (rate of change of f with respect to x or y (holding the other constant) at the point (a,b)) notations for partial derivaties ✔Ans - fx(x,y) = f1(x.y) = ∂z/∂x = ∂f/∂x =zx =Dxf(x,y) = D1f(x,y) locally linear/ locally differentiable ✔Ans - a function z = f(x,y) is locally linear or differentiable at a point (a,b) if the function restricted to progressively smaller neighborhoods of (a,b) becomes progressively closer to a linear function equation of a tangent plane ✔Ans - z =f(a,b) + fx(a,b)(x-a) + fy(a,b) (y-b) dz = fx(a,b)dx + fy(a,b)dy directional derivative of f in direction u at point P ✔Ans - Instantaneous rate of change of f at P with respect to distance traveled from P in the direction u fu(a,b) = Duf(a,b) = lim h--> 0 deltaf/h = lim h--> 0 f(a+fu1,b+hu2) -f(a,b)/h gradient vector of differentiable function f(x,y) at point (a,b) ✔Ans - vector of partial derivatives grad f(a,b) = ∇f(a,b) = fx(a,b)i + fy(a,b)j points in direction of greatest rate of change of f. (usually perpendicular to level curves) the negative is the direction of minimum rate of change and the magnitude is that rate of change directional derivative using gradient and unit vector u ✔Ans - fu(a,b) = fx(a,b)u1 + fy(a,b)u2 = ∇f(a,b) • u chain rule (what is dz/dt if f,g,h are differentiable and z = f(x,y) where x = g(t) and y = h(t)?) ✔Ans - dz/dt = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt) = fx(g(t),h(t))g'(t) + fy(g(t),h(t))h'(t) To find this: draw a diagram expressing the relationship between the variables and label each link in the diagram with the derivative relating the variables at its end. For each path multiple together the derivatives from each step along the path. add the contributions from each path.
nth order partial derivative ✔Ans - computed by taking in sequence n first order partial derivatives of the function mixed partial derivatives ✔Ans - when any of the partial derivatives in a sequence are with respect to different values, then the nth order partial is called mixed. ex. fxy = fx(y) = (∂/∂y)(∂z/∂x) = ∂^2z/∂y∂x (for last one derivatives are taken in the opposite direction first x and then y) fxy = rate of change of x in y direction If fxy and fyx are continuous at (x,y) and (x,y) is not on the boundary of their domain then... ✔Ans - fxy(x,y) = fyx(x,y) Linear and quadratic approximations of function f(x,y) at the point (x,y) = (a,b) ✔Ans - Taylor polynomials of degrees 1 and 2 in two variables. L(x,y) = f(a,b) + fx(a,b)(x-a) + fy(a,b)(y-b) Q(x,y) = f(a,b) + fx(a,b)(x-a) + fy(a,b)(y-b) + (1/2)(fxx(a,b)(x-a)^2 + 2fxy(a,b)(x- a)(y-b) + fyy(a,b)(y-b)^2) local max/min, local extreme ✔Ans - f has a local maximum at point Po if f(Po)>= f(P) for all points P near Po. It has a local minimum at Po if f(Po) <=f(P) for all points P near Po. (both are local extremes) critical points/ stationary points ✔Ans - critical: points where the gradient of f is either 0 or undefined. stationary: critical points where gradient of f = 0. (if f has a local extreme at Po then that is a critical point, if it is differentiable at Po then it is a stationary point) saddle point ✔Ans - neither local max nor local min (like the center of a saddle) Discriminant ✔Ans - fxx(xo,yo)fyy(xo,yo) - (fxy(xo,yo))^ like cross product of fxx fxy fxy fyy If D>0 and fxx > 0 the f has a local minimum at (x0,y0) If D>0 and fxx < 0 the f has a local maximum at (xo,yo) If D<0 then f has a saddle point at (x0,y0) If D = 0, then the test gives you no information
Mass as an integral ✔Ans - ∫ over R of densitydA average value of f over R ✔Ans - (1/area of R)∫ (over R) fdA triple integral ✔Ans - defined by riemann sums of dimensions ∆x by ∆y by ∆z where volume ∆V = ∆x∆y∆z. ∫ (over R) fdV = lim as ∆x,∆y,∆z --> 0 ∑ f(u-ijk, v-ijk, w-ijk)∆x∆y∆z. Can be computed as three consecutive nested integrals. ∫∫∫ f(x,y,z) dz dy dx order of integration chosen according to convenience position vector of point P ✔Ans - a vector whose tail is anchored at the origin and whose head is pointing to the point P. Used to describe the location of a point. If P has coordinates (a,b) then it has a position vector r = ai* + bj parameterized curves ✔Ans - a curve in 2-space can be parameterized by equations of the form x = f(t), y = g(t). As parameter t changes, the point (x,y) traces out a curve. Coordinate representation: x = f(t), y = g(t) Vector representation: r(t) = f(t)i + g(t)j* Or in 3-space r(t) = f(t)i + g(t)j* + h(t)k where z=h(t) parameterizing lines ✔Ans - a line passing through a point P = (xo,yo,zo) and parallel to the vector v = ai + bj + ck has parametric representation (in coordinate and vector form): x = x0 + at, y = y0 + bt, z = z0 + ct and r(t) = r0 + vt where r0 = x0i +y0j + z0k is the position vector of the point P velocity vector ✔Ans - a vector v whose magnitude is the speed of the object and whose direction points in the direction in which the object is moving ||v|| = speed and v is tangent to the objects path. v(t) = r'(t) = dr/dr = lim ∆t-->0 ∆r/∆t = lim ∆t-->0 r(t-∆t) - r(t)/∆t acceleration vector ✔Ans - a vector a(t) if a moving object with position vector r(t) at time t such that a(t) = v'(t) = r''(t)
Uniform circular motion ✔Ans - r(t) = Rcos( ωt)i + Rsin( ωt)j Arc length ✔Ans - If r(t) describes the motion of a point during time period a<=t<=b, then the distance traveled is ∫ (from a to b) ||v(t)||dt If the particle never stops or reverses direction this distance is the length of the curve described by r(t) vector field ✔Ans - a function whose inputs are points in space and whose outputs are vectors. gradient field ✔Ans - when a vector field F(x,y) is equal to the gradient of a scalar function f(x,y). (If there is a scalar function f(x,y) for which ∇f(x,y) = F(x,y) (fxy must equal fyx mixed partials must match) oriented curve ✔Ans - a curve with a particular specified direction of motion along the curve line integral ✔Ans - given a vector field F and an oriented curve C from point P to point Q, the line integral is defined as follows: -divide the curve C into n small arc with endpoints P = r0,r1...rn = Q. -approximate the n arcs with n vectors: ∆ri = r-i+1-ri , i = 1,2,...,n -each point ri has a corresponding vector F(ri) at that point from the vector field that we use to compute the dot product F(ri) • ∆ri -The sum of all of the dot products is the Reimann sum ∑ (i=0 to n-1) F(ri) • ∆ri for the line integral -Then by taking the limit of that summation as ||∆ri || --> which equals ∫(over C) F• dr how to compute a line integral ✔Ans - if you are given a vector field F and an oriented curve C, if you parameterize C with a smooth parameterization r'(t) for a<=t<=b then ∫(over C) F• dr = ∫(from a to b) F(r(t)) • r'(t)dt circulation ✔Ans - When a vector function F is integrated around an oriented closed curve C, then the line integral of F around C is a circulation and is written ∮(over C) F • dr.
∫ (over C) F • dr= ∫ (over R) (∂Q/∂x - ∂P/∂y)dA polar coordinates ✔Ans - x = rcos( ), y= rsin( ), r^2 = x^2 + y^2 , tan( )=y/xθ θ θ double integral in polar coordinates ✔Ans - if R is the region defined by a≤r≤b α θ β ≤ ≤ (with no overlap), then the double integral of f(x,y) over the region R can be expressed by the following iterated integral in polar coordinates: ∫ (over R) f dA =∫ (from α to β) ∫ (from a to b) f(rcos( ),rsin( ))rdrd θ θ θ(thus dA =rdrd )θ
