Equations
The formula that gets added to both sides of an equation to complete the square - sample equation: ax2+bx+c=0
(b/2)2
This operation between two vectors is 0 only if the vectors are orthogonal
dot product
This is the equation used to represent y when changing to cylindrical coordinates
y=r*sin(theta)
The double integral of sqrt(fx2+fy2+1)dA over some region R represents this of a surface S
This is the equation that represents the chain rule with respect to t as applied to the function f(x,y) where x and y are functions of t
df/dt = df/dx*dx/dt+df/dy*dy/dt
The linear approximation formula of f at some point (a,b)
What is L(x,y)=f(a,b)+fx(a,b)*(x-a)+fy(a,b)*(y-b)?
This equation calculates the gradient vector for some function f in 3D space (assuming f has continuous partial derivatives)
gradient of f = fxi+fyj+fzk
This equation represents the z component when changing to spherical coordinates
z=p*cos(theta)
If I'm finding the volume of a partial sphere in spherical coordinates, the integral bounds should have this many variables
0 - the bounds will all be constants because it's a sphere in spherical coordinates
This 3D surface can be represented with the equation (x-h)2+(y-k)2+(z-l)2=r2
A sphere with center (h,k,l)
The equation for the tangent plane to f at some point (a,b,c). [Where f has continuous partial derivatives at the point].
What is z-c=fx(a,b)*(x-a)+fy(a,b)*(y-b)?
In 3D space, the volume of a parallelepiped spanned by vectors a, b, and c can be represented with this expression
det(a,b,c) = a-dot-(bXc)=triple product of a, b, and c
[2 parts] If finding the volume of a cylinder in rectangular coordinates (x,y,z), the inner-most integral should have this many variables in the bounds while the outer-most integral should have this many variables
inner-most integral should have 2 variables
outer-most integral should have 0 variables
This equation is used to check if a field F is conservative (aka checking is F is a gradient field) for F=Pi+Qj
dP/dy=dQ/dx
This is the direction vector of a line L: x=2-t, y=1-t, z=t
v=<-1,-1,2>
This equation is the plane that passes through a point P=(p1,p2,p3) with a normal n=<1,2,3>
What is 1(x-p1)+2(y-p2)+3(z-p3)=0?
A vector field F is a gradient field if there is a potential for F, or a scalar function f, such that this equation is true
F=gradient of f
When finding the volume of an ice cream cone (a cone with part of a sphere on top), this is the upper and lower bounds for p (rho).
Cone: z=sqrt(x2+y2)
Sphere: x2+y2+z2=a2
0<p<a - between 0 and a
When setting up a volume integral in spherical coordinates, this expression represents the volume element, or, the expression that equals dV in a triple integral
dV = p2sin(phi) dp d(theta) d(phi)
any order of integration
This is the partial derivative with respect to y of the function f(x,y)=xcos(x)+y3+ysin(x)
fy=3y2+sin(x)
This expression completes the following equation according to Green's Theorem over some curve C that's the boundary for a region D:
integral(P dx + Q dy) =
= double integral(Qx-Py)dA
over the region D
This expression represents the unit normal vector N to a position vector r(t)
N = (dT/ds) / (|dT/ds|) = N(t) = T'(t)/|T'(t)|
Let D be the part of a disk with radius 2 centered at the origin that lies in the first quadrant. Using polar coordinates, these are the bounds for the variable theta.
0 < theta < pi/2
This expression represents the curl of a vector field F
curl F = dQ/dx-dP/dy
The divergence of a vector field F=Pi+Qj+Rk can be computed using this equation
What is div F = dP/dx+dQ/dy+dR/dz