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Calculus 3 formula sheet

19 formulas Ā· what each one means Ā· a free lesson for each

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Vectors in space

Distance
d = āˆš(x2 āˆ’ x1)2 + (y2 āˆ’ y1)2 + (z2 āˆ’ z1)2
Lesson: Points and distance in space ā†’
Magnitude
ā€–vā€– = āˆša2 + b2 + c2
Lesson: Vectors in space ā†’
Dot product
u ā€¢ v = u1 v1 + u2 v2 + u3 v3
Lesson: Dot product and angle in space ā†’
Angle
cos Īø = u ā€¢ vā€–uā€– ā€–vā€–
Lesson: Dot product and angle in space ā†’
Cross product
u Ɨ v = ijku1u2u3v1v2v3
A vector at 90Ā° to both. Its length is the parallelogram's area.Lesson: Cross product and area ā†’
Line
x = x0 + at, y = y0 + bt, z = z0 + ct
Through (xā‚€, yā‚€, zā‚€) with direction āŸØa, b, cāŸ©.Lesson: Lines in space ā†’
Plane
a(x āˆ’ x0) + b(y āˆ’ y0) + c(z āˆ’ z0) = 0
āŸØa, b, cāŸ© is the normal (at 90Ā° to the plane).Lesson: Planes in space ā†’

Partial derivatives

Partial derivative
fx = āˆ‚fāˆ‚x
Differentiate in x; y stays fixed like a number.Lesson: Partial derivatives ā†’
Tangent plane
z = f(a, b) + fx(a, b)(x āˆ’ a) + fy(a, b)(y āˆ’ b)
Lesson: Tangent planes and linear approximation ā†’
Chain rule
dzdt = āˆ‚zāˆ‚xdxdt + āˆ‚zāˆ‚ydydt
Lesson: Chain rule with several variables ā†’
Gradient
āˆ‡f = āŸØfx, fyāŸ©
Points the way f grows fastest.Lesson: Gradient and directional derivative ā†’
Directional derivative
Du f = āˆ‡f ā€¢ u
u is a unit vector.Lesson: Gradient and directional derivative ā†’
Second derivative test
D = fxx fyy āˆ’ (fxy)2
D > 0 and f_xx > 0: min; D > 0 and f_xx < 0: max; D < 0: saddle.Lesson: Local maximum, minimum and saddle points ā†’
Lagrange multipliers
āˆ‡f = Ī»āˆ‡g, g(x, y) = k
Max or min of f along the curve g = k.Lesson: Lagrange multipliers ā†’

Multiple integrals

Double integral
āˆ¬R f dA = āˆ«baāˆ«dc f(x, y) dy dx
Inside integral first.Lesson: Double integrals over rectangles ā†’
Polar double integral
dA = r dr dĪø
Lesson: Double integrals in polar form ā†’
Triple integral
āˆ­E f dV
Lesson: Triple integrals ā†’
Work (line integral)
W = āˆ«C F ā€¢ dr
Lesson: Line integrals and work ā†’
Green's theorem
āˆ®C P dx + Q dy = āˆ¬R (āˆ‚Qāˆ‚x āˆ’ āˆ‚Pāˆ‚y) dA
Lesson: Green's theorem ā†’

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