## Vector calculus summary

Advanced Calculus Demystified – David Bachman

**Level curves **of a function allow exploration by holding one variable constant (essentially contour plots of a surface). A function has a **limit **at (a,b) only when it approaches the same value no matter how (x,y) approach (a,b). A function is **continuous **at (a,b) if its limit at (a,b) equals the value of the function there.

Functions don’t exist in their domain when 1) there is division by zero, 2) the square root of a negative, 3) the log of a non-positive, or 4) the tan of odd multiples of .

The multivariable calculus version of the chain rule can be given by

**Cylindrical coordinates**

**Spherical coordinates**

In parametrizations, always specify the ranges.

The **vector dot product **is given by

Critical points in f are where . The second derivative is actually four derivatives, best organized in a matrix

The determinant of that matrix at where is .

if then is a local minimum. If then a local maximum.

It is customary to denote the points around the boundary of a domain as (not to be confused with a partial derivative).

The method of **Lagrange multipliers** allows determination of extrema on . Let be defined over and . Find a new function G(x,y) for which is a level curve. is always perpendicular to g’s level curves. $latex\nabla f(x_0,y_0) $ must be parallel to and so one must be a scalar multiple of the other (by ). Those points that satisfy the relation must be included in the evaluation along with the normal critical points.

If and then is the area of a parallelogram. The 3D equivalent is the volume of a parallelepiped. The determinant of the cross product is the area of a parallelogram.

**Line integral**

**Surface integral**

**3D volume**

**Path independence**

**Green’s Theorem**

**Stoke’s Theorem**

**Gauss’ Theorem**

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