Multivariate Calculus 1

Probably more accurately, it’s multidimensional calculus; Analysis of functions of several variables

Represent points in 3d, 3d graphs

Scalar valued functions

\[f(x, y) = x^2 + y^2\] \[z = cos(x)sin(y)\]

Slicing up graphs (useful in partial derivatives)

Contour plots (also known as level plots) is a way to show three-dimensional surface on a two-dimensional plane. It graphs two predicator variables X Y on the y-axis and a response variable Z as contours.

Vector valued functions

Parametric curves

\[f(t) = \begin{bmatrix} tcos(t) \\ tsin(t) \end{bmatrix}\]

Vector fields (fluid flow is an example)

Draw a point, say (1, 2), then from this point, find the resultant vector. Do this for all point on graph and you will see the vector field. Like so:

Parameter vs variable vs constants

  • A constant is a number that doesn’t change as variables change, like \(\pi\).
  • A parameter is a constant that defines a class of equations.
  • A variable is an element of the domain or codomain of a relation.

All parameters are constants but not all constants are parameters. Parameters are typically represented by t or lambda \(\lambda\).

Example, given: \(f: R \to \frac{R}{f(x)} = x^2\)

How to parametrize? What is being asked is representing x and y in function of t, not y in function of x.

  1. \[x = t\]
  2. \[y = t^2\]

3d vector field

Transformations (single variable input, single variable output; one-dimensional input, two-dimensional output; etc.)

Basically, five ways to visualize: traditional graphs, contour maps, parametric curves/surfaces, vector fields, transformations

Partial derivatives, notation is a bit different. It has a \(\frac{\partial f}{\partial x}\) instead of \(\frac{df}{dx}\) which lets readers know it’s a partial derivative. It’s read as “the partial derivative of f with respect to x.” It will take care of one derivative and treat the rest as constant.

Example: \(f(x, y) = x^2y + sin(y)\)

  1. Then: \(\frac{\partial f}{\partial x}\) (1, 2), \(\frac{\partial f}{\partial y}\) (1, 2)
  2. This is: \(\frac{\partial f}{\partial x}\) (1, 2) = \(\frac{\partial}{\partial x} (x^2 2+sin(2))\) at x = 1
  3. = 4x + 0 at x = 1
  4. = 4 at x = 1

Formal definition of a partial derivative \(f_x(x, y) = \lim\limits_{h \to 0} \frac{f(x+h, y) - f(x, y)}{h}\) \(f_y(x, y) = \lim\limits_{h \to 0} \frac{f(x, y+h) - f(x, y)}{h}\)

Symmetry of partial derivatives \(\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x}\)

Schwarz’s theorem

Gradient, gradient is the direction of the steepest ascent

\(\nabla\) for gradient of f

\[\nabla f = \begin{bmatrix} (\partial f)/(\partial x) \\ (\partial f)/(\partial y) \end{bmatrix}\]

Directional derivative \(\nabla_{\vec{w}} f = a \frac{\partial f}{\partial x} + b \frac{\partial f}{\partial y}\)

Formal definition of directional derivative \(D_{\vec{u}} f(x, y) = \lim\limits_{h \to 0} \frac{f(x+ah, y+bh) - f(x, y)}{h}\)

Differentiating parametric curve

  • Position vector = \(r(\vec{t})\)
  • Position vector literally describes a point in space
  • \[r(\vec{a})=x(a)\hat{i} + y(a)\hat{j}\]

\(\vec{r}'(t)\) or \(\frac{d\vec{r}}{dt}\)

\(\vec{r}(t) = x(t)\hat{i} + y(t)\hat{j}\), a < t < b

Multivariable chain rule

Formal definition of multivariable chain rule \(\frac{d}{dt} f(x(t), y(t)) = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt}\)

Vector form of multivariable chain rule

\[\frac{d}{dt} f(x(t), y(t)) = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt}\] \[\begin{bmatrix} (\partial f)/(\partial x) \\ (\partial f)/(\partial y) \end{bmatrix} \begin{bmatrix} \frac{dx}{dt} \\ \frac{dy}{dt} \end{bmatrix}\]

= \(\nabla f * \vec{v}'(t)\) \(\nabla f(\vec{v}(t)) * \vec{v}'(t)\)

Directional derivative chain rule

Curvature (uses a bit more linear algebra knowledge here)

\(\kappa\) (Greek letter kappa) = 1/R

Radius of curvature Definition: The curvature of a curve is \(\kappa = \vert \frac{dT}{ds} \vert\) where T is the unit tangent vector.

Sharp turn, small radius, large curvature

Described parametrically, typically: \(\vec{r}(t)\) = \(\begin{bmatrix} x(t) \\ y(t) \end{bmatrix}\)

Know how to take derivative of position vector valued functions, cross products

There are many formulations for curvature

Partial derivative of vector valued functions

Partial derivative of parametric surface

Partial derivative of vector fields

Partial derivative of vector fields, component by component

Divergence \(\nabla * A = \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z}\) = (rate of change of A in x-direction) + (rate of change of A in y-direction) + (rate of change of A in z-direction)

Curl

Laplacian operator

Harmonic functions

Jacobian

Jacobian matrix Local linearity

Jacobian Determinant tells you how much the area will be stretched out

Example of everything learned so far: Let f(x, y) = \(x^2 – xy\) and \(\vec{v}\) = \(\begin{bmatrix} 2 \\ -2 \end{bmatrix}\)

What is \(\nabla \vec{v}\) f(x, y)?

Take the directional derivative of a multivariable function f(x, y) along a vector \(\vec{v}\), take the dot product between the gradient of f and \(\vec{v}\)

The gradient is:

\(\begin{bmatrix} (\partial f)/(\partial x) (x^2-xy) \\ (\partial f)/(\partial y) (x^2-xy) \end{bmatrix}\) = \(\begin{bmatrix} 2x-y \\ -x \end{bmatrix}\)

Then take the dot product: \(\nabla \vec{v}\) f(x, y) = \(\nabla\) f(x, y) \(\vec{v}\)= \(\begin{bmatrix} 2x-y \\ -x \end{bmatrix}\) \(\begin{bmatrix} 2 \\ -1 \end{bmatrix}\) = (2x – y)2 + (-x)(-1) = 5x – 2y

\(\nabla \vec{v}\) f(x, y) = 5x – 2y

Tangent plane

Linear approximation L(x) \(\approx\) f(a) + f’(a)(x-a)

Quadratic approximation

Hessian matrix

Minima and maxima

Local, global

Critical points

Concavity

Saddle point

Second partial derivative test

Constrained optimization

Lagrange multiplier

Lagrangian

Line integral

Can do linear integral over any field (vector, gravitational, electrical, etc.)

Multivariate Calculus 2

  • Optimization techniques
  • Multiple integrals
  • Line integrals
  • Surface integrals
  • Major integrals concerning their applications

Double integral is integral to find volume under a surface.

Flux

Green’s theorem

2D divergence theorem

Stokes’ theorem

Calculus 3 Problems and Solutions