1. Gradient: The gradient of a scalar function f(x, y, z) is denoted by ∇f or grad f and is given by:

∇f = (∂f/∂x)i + (∂f/∂y)j + (∂f/∂z)k

where i, j, and k are the unit vectors in the x, y, and z directions, respectively.

2. Divergence: The divergence of a vector field F(x, y, z) = P(x, y, z)i + Q(x, y, z)j + R(x, y, z)k is denoted by ∇·F or div F and is given by:

∇·F = (∂P/∂x) + (∂Q/∂y) + (∂R/∂z)

3. Curl: The curl of a vector field F(x, y, z) = P(x, y, z)i + Q(x, y, z)j + R(x, y, z)k is denoted by ∇xF or curl F and is given by:

∇xF = (∂R/∂y - ∂Q/∂z)i - (∂R/∂x - ∂P/∂z)j + (∂Q/∂x - ∂P/∂y)k

4. Line Integral: The line integral of a vector field F(x, y, z) along a curve C parametrized by r(t) = x(t)i + y(t)j + z(t)k, where t ranges from a to b, is given by:

∫(C) F·dr = ∫(a, b) F(r(t))·r'(t) dt

5. Surface Integral: The surface integral of a vector field F(x, y, z) over a surface S parametrized by r(u, v) = x(u, v)i + y(u, v)j + z(u, v)k, where (u, v) ranges over a region D, is given by:

∬(S) F·dS = ∬(D) F(r(u, v))·(∂r/∂u x ∂r/∂v) dudv

6. Volume Integral: The volume integral of a scalar function f(x, y, z) over a solid region V is given by:

∭(V) f(x, y, z) dV

7. Stokes' Theorem: The line integral of a vector field F(x, y, z) around a simple, closed, and positively oriented curve C is equal to the surface integral of the curl of F over a surface S bounded by C:

∫(C) F·dr = ∬(S) (∇xF)·dS

8. Gauss's (Divergence) Theorem: The surface integral of a vector field F(x, y, z) over a closed surface S is equal to the volume integral of the divergence of F over the solid region V enclosed by S:

∬(S) F·dS = ∭(V) (∇·F) dV

9. Green's Theorem: The line integral of a vector field F(x, y) = P(x, y)i + Q(x, y)j around a simple, closed, and positively oriented curve C is equal to the double integral of the partial derivatives of P and Q over the region R enclosed by C:

∫(C) F·dr = ∬(R) (∂Q/∂x - ∂P/∂y) dA