def doit(self, **kwargs): from sympy.vector.functions import laplacian return laplacian(self._expr)
def test_del_operator(): # Tests for curl assert delop ^ Vector.zero == Vector.zero assert (delop ^ Vector.zero).doit() == Vector.zero == curl(Vector.zero) assert delop.cross(Vector.zero) == delop ^ Vector.zero assert (delop ^ i).doit() == Vector.zero assert delop.cross(2 * y ** 2 * j, doit=True) == Vector.zero assert delop.cross(2 * y ** 2 * j) == delop ^ 2 * y ** 2 * j v = x * y * z * (i + j + k) assert ( (delop ^ v).doit() == (-x * y + x * z) * i + (x * y - y * z) * j + (-x * z + y * z) * k == curl(v) ) assert delop ^ v == delop.cross(v) assert ( delop.cross(2 * x ** 2 * j) == (Derivative(0, C.y) - Derivative(2 * C.x ** 2, C.z)) * C.i + (-Derivative(0, C.x) + Derivative(0, C.z)) * C.j + (-Derivative(0, C.y) + Derivative(2 * C.x ** 2, C.x)) * C.k ) assert delop.cross(2 * x ** 2 * j, doit=True) == 4 * x * k == curl(2 * x ** 2 * j) # Tests for divergence assert delop & Vector.zero is S.Zero == divergence(Vector.zero) assert (delop & Vector.zero).doit() is S.Zero assert delop.dot(Vector.zero) == delop & Vector.zero assert (delop & i).doit() is S.Zero assert (delop & x ** 2 * i).doit() == 2 * x == divergence(x ** 2 * i) assert delop.dot(v, doit=True) == x * y + y * z + z * x == divergence(v) assert delop & v == delop.dot(v) assert delop.dot(1 / (x * y * z) * (i + j + k), doit=True) == -1 / ( x * y * z ** 2 ) - 1 / (x * y ** 2 * z) - 1 / (x ** 2 * y * z) v = x * i + y * j + z * k assert delop & v == Derivative(C.x, C.x) + Derivative(C.y, C.y) + Derivative( C.z, C.z ) assert delop.dot(v, doit=True) == 3 == divergence(v) assert delop & v == delop.dot(v) assert simplify((delop & v).doit()) == 3 # Tests for gradient assert delop.gradient(0, doit=True) == Vector.zero == gradient(0) assert delop.gradient(0) == delop(0) assert (delop(S.Zero)).doit() == Vector.zero assert ( delop(x) == (Derivative(C.x, C.x)) * C.i + (Derivative(C.x, C.y)) * C.j + (Derivative(C.x, C.z)) * C.k ) assert (delop(x)).doit() == i == gradient(x) assert ( delop(x * y * z) == (Derivative(C.x * C.y * C.z, C.x)) * C.i + (Derivative(C.x * C.y * C.z, C.y)) * C.j + (Derivative(C.x * C.y * C.z, C.z)) * C.k ) assert ( delop.gradient(x * y * z, doit=True) == y * z * i + z * x * j + x * y * k == gradient(x * y * z) ) assert delop(x * y * z) == delop.gradient(x * y * z) assert (delop(2 * x ** 2)).doit() == 4 * x * i assert (delop(a * sin(y) / x)).doit() == -a * sin(y) / x ** 2 * i + a * cos( y ) / x * j # Tests for directional derivative assert (Vector.zero & delop)(a) is S.Zero assert ((Vector.zero & delop)(a)).doit() is S.Zero assert ((v & delop)(Vector.zero)).doit() == Vector.zero assert ((v & delop)(S.Zero)).doit() is S.Zero assert ((i & delop)(x)).doit() == 1 assert ((j & delop)(y)).doit() == 1 assert ((k & delop)(z)).doit() == 1 assert ((i & delop)(x * y * z)).doit() == y * z assert ((v & delop)(x)).doit() == x assert ((v & delop)(x * y * z)).doit() == 3 * x * y * z assert (v & delop)(x + y + z) == C.x + C.y + C.z assert ((v & delop)(x + y + z)).doit() == x + y + z assert ((v & delop)(v)).doit() == v assert ((i & delop)(v)).doit() == i assert ((j & delop)(v)).doit() == j assert ((k & delop)(v)).doit() == k assert ((v & delop)(Vector.zero)).doit() == Vector.zero # Tests for laplacian on scalar fields assert laplacian(x * y * z) is S.Zero assert laplacian(x ** 2) == S(2) assert ( laplacian(x ** 2 * y ** 2 * z ** 2) == 2 * y ** 2 * z ** 2 + 2 * x ** 2 * z ** 2 + 2 * x ** 2 * y ** 2 ) A = CoordSys3D( "A", transformation="spherical", variable_names=["r", "theta", "phi"] ) B = CoordSys3D( "B", transformation="cylindrical", variable_names=["r", "theta", "z"] ) assert laplacian(A.r + A.theta + A.phi) == 2 / A.r + cos(A.theta) / ( A.r ** 2 * sin(A.theta) ) assert laplacian(B.r + B.theta + B.z) == 1 / B.r # Tests for laplacian on vector fields assert laplacian(x * y * z * (i + j + k)) == Vector.zero assert ( laplacian(x * y ** 2 * z * (i + j + k)) == 2 * x * z * i + 2 * x * z * j + 2 * x * z * k )
def test_del_operator(): # Tests for curl assert delop ^ Vector.zero == Vector.zero assert ((delop ^ Vector.zero).doit() == Vector.zero == curl(Vector.zero)) assert delop.cross(Vector.zero) == delop ^ Vector.zero assert (delop ^ i).doit() == Vector.zero assert delop.cross(2*y**2*j, doit=True) == Vector.zero assert delop.cross(2*y**2*j) == delop ^ 2*y**2*j v = x*y*z * (i + j + k) assert ((delop ^ v).doit() == (-x*y + x*z)*i + (x*y - y*z)*j + (-x*z + y*z)*k == curl(v)) assert delop ^ v == delop.cross(v) assert (delop.cross(2*x**2*j) == (Derivative(0, C.y) - Derivative(2*C.x**2, C.z))*C.i + (-Derivative(0, C.x) + Derivative(0, C.z))*C.j + (-Derivative(0, C.y) + Derivative(2*C.x**2, C.x))*C.k) assert (delop.cross(2*x**2*j, doit=True) == 4*x*k == curl(2*x**2*j)) #Tests for divergence assert delop & Vector.zero == S(0) == divergence(Vector.zero) assert (delop & Vector.zero).doit() == S(0) assert delop.dot(Vector.zero) == delop & Vector.zero assert (delop & i).doit() == S(0) assert (delop & x**2*i).doit() == 2*x == divergence(x**2*i) assert (delop.dot(v, doit=True) == x*y + y*z + z*x == divergence(v)) assert delop & v == delop.dot(v) assert delop.dot(1/(x*y*z) * (i + j + k), doit=True) == \ - 1 / (x*y*z**2) - 1 / (x*y**2*z) - 1 / (x**2*y*z) v = x*i + y*j + z*k assert (delop & v == Derivative(C.x, C.x) + Derivative(C.y, C.y) + Derivative(C.z, C.z)) assert delop.dot(v, doit=True) == 3 == divergence(v) assert delop & v == delop.dot(v) assert simplify((delop & v).doit()) == 3 #Tests for gradient assert (delop.gradient(0, doit=True) == Vector.zero == gradient(0)) assert delop.gradient(0) == delop(0) assert (delop(S(0))).doit() == Vector.zero assert (delop(x) == (Derivative(C.x, C.x))*C.i + (Derivative(C.x, C.y))*C.j + (Derivative(C.x, C.z))*C.k) assert (delop(x)).doit() == i == gradient(x) assert (delop(x*y*z) == (Derivative(C.x*C.y*C.z, C.x))*C.i + (Derivative(C.x*C.y*C.z, C.y))*C.j + (Derivative(C.x*C.y*C.z, C.z))*C.k) assert (delop.gradient(x*y*z, doit=True) == y*z*i + z*x*j + x*y*k == gradient(x*y*z)) assert delop(x*y*z) == delop.gradient(x*y*z) assert (delop(2*x**2)).doit() == 4*x*i assert ((delop(a*sin(y) / x)).doit() == -a*sin(y)/x**2 * i + a*cos(y)/x * j) #Tests for directional derivative assert (Vector.zero & delop)(a) == S(0) assert ((Vector.zero & delop)(a)).doit() == S(0) assert ((v & delop)(Vector.zero)).doit() == Vector.zero assert ((v & delop)(S(0))).doit() == S(0) assert ((i & delop)(x)).doit() == 1 assert ((j & delop)(y)).doit() == 1 assert ((k & delop)(z)).doit() == 1 assert ((i & delop)(x*y*z)).doit() == y*z assert ((v & delop)(x)).doit() == x assert ((v & delop)(x*y*z)).doit() == 3*x*y*z assert (v & delop)(x + y + z) == C.x + C.y + C.z assert ((v & delop)(x + y + z)).doit() == x + y + z assert ((v & delop)(v)).doit() == v assert ((i & delop)(v)).doit() == i assert ((j & delop)(v)).doit() == j assert ((k & delop)(v)).doit() == k assert ((v & delop)(Vector.zero)).doit() == Vector.zero # Tests for laplacian on scalar fields assert laplacian(x*y*z) == S.Zero assert laplacian(x**2) == S(2) assert laplacian(x**2*y**2*z**2) == \ 2*y**2*z**2 + 2*x**2*z**2 + 2*x**2*y**2 # Tests for laplacian on vector fields assert laplacian(x*y*z*(i + j + k)) == Vector.zero assert laplacian(x*y**2*z*(i + j + k)) == \ 2*x*z*i + 2*x*z*j + 2*x*z*k
def test_del_operator(): # Tests for curl assert delop ^ Vector.zero == Vector.zero assert ((delop ^ Vector.zero).doit() == Vector.zero == curl(Vector.zero)) assert delop.cross(Vector.zero) == delop ^ Vector.zero assert (delop ^ i).doit() == Vector.zero assert delop.cross(2 * y**2 * j, doit=True) == Vector.zero assert delop.cross(2 * y**2 * j) == delop ^ 2 * y**2 * j v = x * y * z * (i + j + k) assert ((delop ^ v).doit() == (-x * y + x * z) * i + (x * y - y * z) * j + (-x * z + y * z) * k == curl(v)) assert delop ^ v == delop.cross(v) assert (delop.cross( 2 * x**2 * j) == (Derivative(0, C.y) - Derivative(2 * C.x**2, C.z)) * C.i + (-Derivative(0, C.x) + Derivative(0, C.z)) * C.j + (-Derivative(0, C.y) + Derivative(2 * C.x**2, C.x)) * C.k) assert (delop.cross(2 * x**2 * j, doit=True) == 4 * x * k == curl( 2 * x**2 * j)) #Tests for divergence assert delop & Vector.zero == S(0) == divergence(Vector.zero) assert (delop & Vector.zero).doit() == S(0) assert delop.dot(Vector.zero) == delop & Vector.zero assert (delop & i).doit() == S(0) assert (delop & x**2 * i).doit() == 2 * x == divergence(x**2 * i) assert (delop.dot(v, doit=True) == x * y + y * z + z * x == divergence(v)) assert delop & v == delop.dot(v) assert delop.dot(1/(x*y*z) * (i + j + k), doit=True) == \ - 1 / (x*y*z**2) - 1 / (x*y**2*z) - 1 / (x**2*y*z) v = x * i + y * j + z * k assert (delop & v == Derivative(C.x, C.x) + Derivative(C.y, C.y) + Derivative(C.z, C.z)) assert delop.dot(v, doit=True) == 3 == divergence(v) assert delop & v == delop.dot(v) assert simplify((delop & v).doit()) == 3 #Tests for gradient assert (delop.gradient(0, doit=True) == Vector.zero == gradient(0)) assert delop.gradient(0) == delop(0) assert (delop(S(0))).doit() == Vector.zero assert (delop(x) == (Derivative(C.x, C.x)) * C.i + (Derivative(C.x, C.y)) * C.j + (Derivative(C.x, C.z)) * C.k) assert (delop(x)).doit() == i == gradient(x) assert (delop(x * y * z) == (Derivative(C.x * C.y * C.z, C.x)) * C.i + (Derivative(C.x * C.y * C.z, C.y)) * C.j + (Derivative(C.x * C.y * C.z, C.z)) * C.k) assert (delop.gradient(x * y * z, doit=True) == y * z * i + z * x * j + x * y * k == gradient(x * y * z)) assert delop(x * y * z) == delop.gradient(x * y * z) assert (delop(2 * x**2)).doit() == 4 * x * i assert ((delop(a * sin(y) / x)).doit() == -a * sin(y) / x**2 * i + a * cos(y) / x * j) #Tests for directional derivative assert (Vector.zero & delop)(a) == S(0) assert ((Vector.zero & delop)(a)).doit() == S(0) assert ((v & delop)(Vector.zero)).doit() == Vector.zero assert ((v & delop)(S(0))).doit() == S(0) assert ((i & delop)(x)).doit() == 1 assert ((j & delop)(y)).doit() == 1 assert ((k & delop)(z)).doit() == 1 assert ((i & delop)(x * y * z)).doit() == y * z assert ((v & delop)(x)).doit() == x assert ((v & delop)(x * y * z)).doit() == 3 * x * y * z assert (v & delop)(x + y + z) == C.x + C.y + C.z assert ((v & delop)(x + y + z)).doit() == x + y + z assert ((v & delop)(v)).doit() == v assert ((i & delop)(v)).doit() == i assert ((j & delop)(v)).doit() == j assert ((k & delop)(v)).doit() == k assert ((v & delop)(Vector.zero)).doit() == Vector.zero # Tests for laplacian on scalar fields assert laplacian(x * y * z) == S.Zero assert laplacian(x**2) == S(2) assert laplacian(x**2*y**2*z**2) == \ 2*y**2*z**2 + 2*x**2*z**2 + 2*x**2*y**2 # Tests for laplacian on vector fields assert laplacian(x * y * z * (i + j + k)) == Vector.zero assert laplacian(x*y**2*z*(i + j + k)) == \ 2*x*z*i + 2*x*z*j + 2*x*z*k