Derivatives of x and y, both squared, with square root, from a to b

Length of a parametric curve

r as a function of theta and derivative, both squared, with square root, from alpha to beta

Length of a polar curve

1/2 times r as a function of theta squared from alpha to beta

Area of a polar region

dot product over magnitude TIMES the other vector over magnitude

Projection of vector a onto b

r = r0 + tv x = x0 + ta, y = y0 + tb, z = z0 + tc,

where r0 = vector of x0, y0, z0 and v = vector of a, b, c.

where r0 = vector of x0, y0, z0 and v = vector of a, b, c.

Equations of lines

z − z0 = fx(x0, y0)(x − x0) + fy(x0, y0)(y − y0)

The equation of the tangent plane to a graph of f(x, y) at (x0, y0, z0), z0 = f(x0, y0)

div(curl (G)) = 0 all the time

Divergence and Curl #1

x^2 + y^2 = z

Elliptic Paraboloid (rocket cap)

x^2 + y^2 + z^2 = 1

Ellipsoid (football)

x^2 + y^2 = z^2

Cone (both sides like the ice cream cone has a twin coming down his butt)

curl F = 0.

F is conservative IF AND ONLY IF . . .

Gradient x F

curl F

Gradient . F

divergence F

S is oriented, smooth surface BOUNDED by simple closed smooth boundary curve C with positive orientation.

Stokes’ Theorem

Magnitude of r'(t)

Length of curve C

conservative, Py, Qx

If gradient f = vector field F, then ______________________ and ___________ equals ____________ .

Positively oriented, piecewise-smooth, simple closed curve in the plane, let D be the region bounded by C, P and Q has continuous partial derivatives on open region that contains D, then

Green’s Theorem (just like Stokes’ Theorem, but without a z-component)

A times B = Magnitude A times Mag B COS THETA

The Scalar Product

Let E simple solid region, S boundary surface of E, with positive outward orientation. F has CTS partial derivatives on an open region that contains E.

Divergence Theorem

If a smooth parametric surface S is given by the equation r(uv) (and components), and S is covered just once throughout domain D, then surface area S is double integral (area) of magnitude cross-product two vectors.

Surface area of a Surface

position vectors, r, of points on a plane through a point with the position vector r0 a normal n satisfy the equation (r − r0) · n = 0, a(x − x0) + b(y − y0) + c(z − z0) = 0,

Equations of planes