For Existing Users
Building on Mathematica's rich paradigms, Version 10 introduces a host of new ideas that not only extend the domain of Mathematica, but also streamline and make more efficient many existing patterns of usage.
Highlights of the Rest:
Interactive Conveniences
Computation-aware multiple undo »
The problem of multiple undo in Mathematica has been solved!
Visual hinting of computation dependency
When you edit an input line, its corresponding output goes gray to indicate that it's not connected anymore
Hovering help box »
Whenever you type a function, help on it is just one easy click away
Inline natural language input »
Use Ctrl + = to insert entity names and much more in natural language
Access more styles with Tab
Just press Tab before you type content in a cell to go from section to subsection to...
Autocompletion Everywhere
Option value autocompletion »
Context-sensitive autocompletion that suggests detailed settings for options
Instant autocompletion wizards »
When you need to pick a file or a color, there's always a wizard to help
String autocompletion »
Autocomplete millions of entity names, properties, etc. when they appear in functions
Iconic preview autocompletion »
See visual previews of fonts, plot themes, styles, and more
Code/text/string/etc. input autoconversion »
Just start to type, and a heuristic suggests creating a code, text, string, or other cell
New Core Data Structures
Key-value associations »
Introducing <| ... |>: highly efficient key-value associations deeply integrated into the system
It's SQL, it's NoSQL—it's Dataset »
Introducing a new way to handle relational and hierarchical structured data
Entities for knowledge representation »
Countries, chemicals, movies: there are built-in symbolic entities for all of them
Inactivate and manipulate! »
Make any part of any expression completely inert, for formal manipulation, code transformation, etc.
Geometric region objects »
As part of the large-scale geometry system, there's now a uniform way to represent and use any geometric region
Language Conveniences & Functional Programming
Many more currying and operator forms »
Select[criterion][list] is Select[list, criterion]—and lots of other functions have operator forms, too
The "Key" family »
Lots of functions for handling key-value pairs—as well as lists of rules
The "First" family »
FirstPosition, FirstCase, SelectFirst: no need to check and select first parts anymore
Functions that return associations »
PositionIndex, Count, etc.: where, how many, etc., given as associations
The "By" family »
MaximalBy, CountsBy, GroupBy: lots of "adverb-like" forms of functions
Leveraging Association for lists of rules »
When it makes sense, what works for Association also works for lists of rules
New Syntax
<| ... |> — Association »
Key-value associations: for example, <|a->1, b->2|>
#x — named arguments »
#x is Part[#, "x"] to make it easy to use values from an association
@* and /* — Composition and RightComposition »
Making it easier and easier to do serious functional programming
Templates for Everything
Templates for strings »
It's now so easy to generate strings from templates...
Templated document generation »
So much easier than notebook programming: just use templates to create documents
Templates for the web »
Just a few special tags in your HTML, and you've got a powerful computable template
Notebook templates »
As easy as making a notebook—but now it's a template for making other notebooks
Construct and Pluralize »
Write templates with Pluralize to make words plural when necessary
More Time & Place
Symbolic date objects »
Date and time objects allowing arithmetic, interaction with units, etc.
ARCH, GARCH, and friends »
Lots of powerful time series analysis, modeling, and forecasting
Here, Now, Tomorrow, etc. »
It's so easy to read code when it simply says "Here" or "Now"
Rise, set, etc. for Sun, Moon »
Convenient immediate astronomical computation
Geoposition objects »
There's now a built-in way to represent every point on the Earth (or Moon etc.)
"What time is it?" anywhere »
LocalTime and new time zone handling
Time and event series »
Time series are now part of the core of the system
Geo everything »
Geo computations—and geo maps of everywhere on the Earth—and beyond
Yet More Pure & Applied Math
Additional special functions »
Four more Dirichlet functions, a couple of Fresnels... not to mention Farey sequences or fibonorials
Finite elements! »
Finite elements have arrived—after nearly a decade of development
Mandelbrot and Julia sets »
Efficient functions for computing and visualizing Mandelbrot and Julia sets
Hidden Markov models »
Another piece in the whole framework of random processes and modeling
Math Conveniences
DSolveValue and friends »
Get just the main result from solving an equation...
Matrix predicates »
Test if matrices are orthogonal, positive semidefinite, antihermitean, etc.
Everything over regions »
Solve, Reduce, Integrate, Optimize, etc. all now work with arbitrary geometric regions
Carry units when you do calculus »
DSolve, Integrate, Interpolate, and many more now handle quantities with units
Sigmoid, PowerRange, ...
Yet more functions to make mathematical code convenient and readable
For Math Classes & More
Display math unevaluated »
Use Inactive to maintain inactive versions of math expressions, for manipulation, display, etc.
All the plane and solid geometry you need »
Areas, volumes, distances to figures, etc.—all built in and symbolic
Function domains and ranges »
Full symbolic representations of domains and ranges (and periods) for all sorts of functions
Manipulate implicit math »
Differentiate a formal sum, operate under an integral—do formal operations on unevaluated math
Curve computations »
Curvature, arc length, Frenet, and more
Triangle math »
Use SASTriangle etc. to specify a triangle, then solve for anything about it
Number-line plots »
Beyond 2D and 3D plotting; this is 1D plotting—of points, intervals, etc.
Bigger, Faster, Better Computations
Faster bignums, linear algebra, and more
New optimizations for new machine architectures, new algorithms, etc.
1,000x speed-up in polynomial reduction »
Reduce and solve systems of exact polynomial equations much faster
Lots more sums and products »
New "functional telescoping" algorithms—significantly expanding the universe of doable sums and products
100,000+ numerical solutions »
A major new algorithm for solving numerical systems of polynomial equations
Faster nearest neighbors »
A new algorithm finds nearest neighbors faster in high dimensions
Reduced memory usage
Optimizations in several core data structures reduce memory usage by factors of 2 or more
Breaking the complexity barrier for TSP »
Traveling Salesman Problems with 1,000+ points are now easy
Hybrid symbolic differential equations »
For the first time ever: symbolically solve ODEs with delays and discrete events
Shared library optimization
The kernel now runs as a shared library, cutting the code space needed to run multiple kernels
Elegant New Looks—and Capabilities
A fresh look for visualization »
Completely redesigned and modernized look for visualization
Get back the classic look... »
Just set a plot theme to get back the classic look for all visualizations
Colors display as swatches »
Now you can immediately see what RGBColor[1,.7,.8] is: it just displays as a swatch
Legends for any visualization »
Automated algorithms for creating legends, including quantities and units if needed
Visual summaries of symbolic objects
InterpolatingFunction, NearestFunction, InputStream, etc. now display in a consistent format with drill-down
Plot themes: one option to pick everything »
Minimal, web, business, scientific, sparkline...: pick a theme to automatically set graphics options
Multicolumn output »
Automatically format anything into multiple (perhaps ragged) columns
A few new characters... »
... and of course, 
Interacting with the Outside World
Interpreters for everything »
500+ built-in string interpreters—for everything from cities to cat breeds
URL manipulation »
Encoding, decoding, shortening, calling like functions, etc.: all kinds of URL operations
Touch-device support »
TouchPosition for touchscreens
Two-way interaction with external processes »
Start an external process, and do repeated I/O with it
Sending tweets, etc. »
Tweet, update status, etc. from inside any program
More sophisticated DLL communication »
Send images, sparse arrays, and more through LibraryLink to DLLs, and get called back too
Simple string I/O gets simpler »
ReadString, ReadLine, etc.: exactly what you need for low-level string I/O
More robust image capture »
Better support for the zoo of camera connections that exist on computers
Modernized SQL connectivity »
New drivers and more streamlining come to DatabaseLink
Questions? Comments? Contact a Wolfram expert »
