Calculus three lecture notes PDF

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Calculus three lecture notes...

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curl f - meaningless; curl only for vector fields grad f - vector field div F - scalar field grad F - meaningless; gradient only for scalar fields div f - meaningless; divergence of scalar field can't be taken curl F - vector field divergence definition - del * F; Px + Qy + Rz curl definition - del x F; gradient definition - del * f; f Chain Rule - dz/dt = df/dx Dx/Dt + df/dy Dy/Dt Zx - -Fx/Fz Zy - -Fy/Fz Equation of Tangent Plane to Surface at a point - Z-Zo=fx(xo,yo)(x-xo) + fy(xo,yo)(y-yo) r^2 = - x^2+y^2 SA = - 2pirl L - sqrt(fx^2 + fy^2) ; sqrt(r^2 + dr/d0^2)d0 parametric equation to tangent line to the given parametric equations and point - find r't, r(t of given point), + t curvature - how fast unit tan changes w.r.t. arclength T(t) - r'(t)/|r'(t)|

K - |T'(t)/r'(t)| N(t) normal vector - T'(t)/|T'(t)| level curves - set f(x,y)=z equal to "c" - height, solve, graph. Equation of a plane - A(x-xo) + B(y-yo) + C(z-zo) = 0 places where max/min can occur - endpoints, d/dx = 0, dne ; Stationary points, singular points, boundaries, critical points of f inside D, f(x, ) f( ,y) Find area by rectangles - f(x,y)^A + sigma f(xbar,ybar) ^A Find area of region bounded by two graphs - int(larger - smaller) dx ; int(a,b)int(bottom,top) f dxdy ; int(a,b) int(left,right) f dxdy dA in polars - rdrd0 sin(phi) - r/rho cos(phi) - z/p x in spherical - rhosinphicos0 y in spherical - rhosinphisin0 rho - sqrt(r^2+z^2) z in spherical - rhocosphi dV in spherical - rho^2 sinphi drho d0 dphi int c f(x,y)dS - int(a,b) f(x(t),y(t)) sqrt( dx/dt^2 + dy/dt^2) dt a Force F = applied to move an object from point 1 to point 2, w=? - int c F*dr = int(t1,t2) * jacobian - partial(x,y)/partial(u,v) = f(x,y) X f(u,v) = |xu,yu / xv, yv|

Clockwise - negative conservative methods - px=qy or curl F = 0 SA of anything - doubleintegral dS...