Implementing Kronecker delta functions

MJxPrep.js

@@ -90,6 +90,14 @@ if (window.MJxPrep) {
     eqn = wrapVariables(eqn)
     eqn = wrapFuncCalls(eqn)
 
+    // Fix Kronecker deltas now: kronecker(a, b) -> kronecker_{a, b}
+    // The wrapping interferes with parsing of delta_{complexthings},
+    // due to not satisfying the usual regex forms for a variable or function
+    eqn = replaceFunctionCalls(eqn, 'kronecker', function(funcName, args) {
+      if (args.length !== 2) {return funcToSelf(funcName, args) ;}
+      return 'delta_{' + args[0] + ',' + args[1] + '}'
+    })
+
     // Return the preprocessed equation
     return eqn;
   }

course/course/course.xml

@@ -28,6 +28,7 @@
         <problem url_name="formula8"/>
         <problem url_name="formula9"/>
         <problem url_name="formula10"/>
+        <problem url_name="formula11"/>
       </vertical>
     </sequential>
     <sequential url_name="matrixgrader" display_name="MatrixGrader">

course/problem/formula11.xml

@@ -0,0 +1,49 @@
+<problem display_name="Kronecker Deltas" showanswer="always" weight="10" attempts="">
+
+  <p>
+    The Kronecker delta often appears in expressions, and is included in the library. To invoke it, use <code>kronecker(x, y)</code>. Note that when using the AsciiMath preprocessor, this will render correctly. We decided against calling it <code>delta(x, y)</code>, as [mathjaxinline]\delta[/mathjaxinline] is a common user function name, and we do not support name overloading of functions.
+  </p>
+
+  <p>
+    When using a Kronecker delta in a solution, we strongly recommend using integer sampling in a small range (e.g., 1 to 4) and using many samples (e.g., 30) so that the various possible permutations are triggered in the grading.
+  </p>
+
+  <p>
+    What is the derivative [mathjaxinline]\frac{\partial x_i}{\partial x_j}[/mathjaxinline]?
+  </p>
+
+  <p>Suggested inputs:</p>
+  <ul>
+    <li><code>kronecker(i, j)</code> is the correct input</li>
+    <li><code>kronecker(i+1, j+1)</code> is also correct</li>
+    <li><code>kronecker(i+1, j)</code> is incorrect</li>
+    <li><code>1</code> and <code>0</code> are incorrect, but may be graded as correct in very rare circumstances, depending on the ranges and number of samples used</li>
+  </ul>
+
+<script type="text/python" system_path="python_lib">
+from mitxgraders import *
+grader = FormulaGrader(
+    answers='kronecker(i, j)',
+    variables=['i', 'j'],
+    sample_from={
+        'i': IntegerRange([1, 4]),
+        'j': IntegerRange([1, 4])
+    },
+    samples=30,
+    suppress_warnings=True  # Needed because we're using i and j as variables
+)
+</script>
+
+<p style="display:inline">[mathjaxinline]\displaystyle \frac{\partial x_i}{\partial x_j} =[/mathjaxinline] </p>
+<customresponse cfn="grader" inline="1">
+  <textline math="1" inline="1" correct_answer="x^n/fact(n)" preprocessorClassName="MJxPrep" preprocessorSrc="/static/MJxPrep.js"/>
+</customresponse>
+
+  <h3>Resources</h3>
+  <ul>
+    <li>
+      <a href="https://github.com/mitodl/mitx-grading-library/tree/master/course/problem/formula11.xml" target="_blank">View source</a>
+    </li>
+  </ul>
+
+</problem>

docs/grading_math/functions_and_constants.md

@@ -43,6 +43,7 @@ By default, all of the following functions are made available to students in `Fo
 - `re(x)` Real part of a complex expression*
 - `im(x)` Imaginary part of a complex expression*
 - `conj(x)` Complex conjugate of a complex expression*
+- `kronecker(x, y)` Kronecker delta* (Note that we highly recommend integer sampling over a short range (eg, 1 to 4) when kronecker deltas appear in an answer, and using many samples (eg, 30) so that most permutations appear in the sampling.)
 
 ## MatrixGrader Default Functions
 

mitxgraders-js/MJxPrep.test.js

@@ -108,6 +108,11 @@ describe('preProcessEqn', () => {
     expect(preProcessEqn(eqn)).toBe('{:cross({:x:}):}+{:({:a:}+{:b:}) {:times:} {:c:}:}+{:y:}')
   } )
 
+  it('replaces Kronecker delta functions', () => {
+    const eqn = 'kronecker(x) + kronecker(a + b, c)'
+    expect(preProcessEqn(eqn)).toBe('{:kronecker({:x:}):}+{:delta_{{:a:}+{:b:},{:c:}}:}')
+  } )
+
   it('wraps variables and function calls', () => {
     // The behavior of wrapVariables and wrapFunCalls is primarily described
     // in their dedicated tests

mitxgraders/helpers/calc/mathfuncs.py

@@ -88,6 +88,18 @@ def arctan2(x, y):
 
     return np.arctan2(y, x)
 
+# NOTE: tests are in a separate file, NOT doctests.
+# see https://bugs.python.org/issue6835
+@SpecifyDomain.make_decorator((1,), (1,))
+def kronecker(x, y):
+    """
+    Returns 1 if x==y, and 0 otherwise.
+    Note that this should really only be used for integer expressions.
+    """
+    if x == y:
+        return 1
+    return 0
+
 def content_if_0d_array(obj):
     """
     If obj is a 0d numpy array, return its contents. Otherwise, return item.
@@ -261,6 +273,7 @@ def has_one_scalar_input(display_name):
                     for key in ELEMENTWISE_FUNCTIONS}
 
 SCALAR_FUNCTIONS['arctan2'] = arctan2
+SCALAR_FUNCTIONS['kronecker'] = kronecker
 
 ARRAY_FUNCTIONS = {
     're': real,

tests/formulagrader/test_formulagrader.py

@@ -531,6 +531,7 @@ def test_fg_debug_log():
     "    'fact': <function factorial at 0x...>,<br/>\n"
     "    'factorial': <function factorial at 0x...>,<br/>\n"
     "    'im': <function imag at 0x...>,<br/>\n"
+    "    'kronecker': <function kronecker at 0x...>,<br/>\n"
     "    'ln': <function log at 0x...>,<br/>\n"
     "    'log10': <function log10 at 0x...>,<br/>\n"
     "    'log2': <function log2 at 0x...>,<br/>\n"

tests/helpers/calc/test_mathfuncs.py

@@ -4,7 +4,7 @@
 from pytest import approx, raises
 from mitxgraders.helpers.calc.exceptions import DomainError, FunctionEvalError
 from mitxgraders.helpers.calc.mathfuncs import (
-    arctan2,
+    arctan2, kronecker,
     cot, arccot,
     csc, arccsc,
     sec, arcsec,
@@ -125,3 +125,10 @@ def test_arctan2():
     match = 'arctan2\(0, 0\) is undefined'
     with raises(FunctionEvalError, match=match):
         arctan2(0, 0)
+
+def test_kronecker():
+    assert kronecker(1, 1) == 1
+    assert kronecker(1, 0) == 0
+    assert kronecker(0, 1) == 0
+    assert kronecker(0, 0) == 1
+    assert kronecker(1.5, 1.5) == 1