Final Math 53 Multivariable Calculus Study Guide Zworski Spring 2017 UC Berkeley

Length of a parametric curve
Derivatives of x and y, both squared, with square root, from a to b
Length of a parametric curve
Length of a polar curve
r as a function of theta and derivative, both squared, with square root, from alpha to beta
Length of a polar curve
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1/2 times r as a function of theta squared from alpha to beta
Area of a polar region
Projection of vector a onto b
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.
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
Elliptic Paraboloid (rocket cap)
x^2 + y^2 = z
Elliptic Paraboloid (rocket cap)
Ellipsoid (football)
x^2 + y^2 + z^2 = 1
Ellipsoid (football)
Cone (both sides like the ice cream cone has a twin coming down his butt)
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
Stokes' Theorem
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 ____________ .
Green's Theorem (just like Stokes' Theorem, but without a z-component)
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)
The Scalar Product
A times B = Magnitude A times Mag B COS THETA
The Scalar Product
Divergence Theorem
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
Equations of planes
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
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