QUICK REVIEW OF INTEGRAL VECTOR CALCULUS -------------Warren D Smith April 2003------- A VOLUME integral of a scalar F(x,y,z) is a triple integral F(x,y,z) dxdydz. dxdydz is sometimes abbreviated "dVolume". The bounds on the integrals have to be chosen to describe the region we are integrating over. For example a 3x5x7 box might be x = 0 to 3 y = 0 to 5 z = 1 to 8 and a cylinder of radius 2 and height 3 might be z = 1 to 4 <-- height 3 innermost integral 2 2 x = -SquareRoot(4-y ) to +SquareRoot(4-y ) y = -2 to 2 outermost integral (choose the order -- inner to outer -- of the integrals to make the problem easiest and also so that if one integral's bounds depend on a variable (here the x bounds depend on y) that variable had better be integrated over OUTSIDE the integral with those bounds (dy is outside of dx). In 2D, you can make a double integral over an area instead of a triple integral over a volume. (Same thing, just 2 not 3, and darea not dvolume.) To CHANGE VARIABLES in a volume integral from x,y,z to a,b,c, where there are some known relations between a,b,c and x,y,z (so that, given a point x,y,z you have a way to determine what a,b,c it is, and vice versa) one uses JacobianDeterminant dx dy dz = Volume Adjustment * da db dc factor where the jacobian matrix is: dx dx dx -- -- -- da db dc dy dy dy -- -- -- da db dc dz dz dz -- -- -- da db dc and you also have to change the bounds on the integrals appropriately so that the set of a,b,c's corresponds to the same point-set described by x,y,z coordinates. Sometimes doing some integral is a lot easier in a nice frendly coordinate system than it is in cartesian coordinates. IMPORTANT COORDINATE SYSTEMS: cartesian coordinates x,y,z in 3D: x = x y = y z = z dx dy dz = dx dy dz <-- volume adjust factor (Jacobian det) = 1 polar coordinates r,theta in 2D: x = r cos theta y = r sin theta dx dy = r dr dtheta <-- area adjust factor (Jacobian det) = r cylindrical coordinates r, theta, z in 3D: x = r cos theta y = r sin theta z = z dx dy dz = r dr dtheta dz <-- volume adjust factor (Jacobian det) = r spherical coordinates r, theta, phi in 3D: x = r cos theta sin phi y = r sin theta sin phi z = r cos phi 2 2 dx dy dz = r sin(phi) dr dtheta dphi <-- volume adjust factor (Jacobian det) = r sin(phi) A SURFACE integral OF A SCALAR (which is a double integral since a surface is TWO dimensional) is an integral / / | | F(x,y,z) . dAreaScalar = integral of F over the surface / / over a surface where F is a scalar-valued function where: (1) the surface has to be parameterized, i.e. (x,y,z) = R(p,q) where p and q are the two parameters and R is a VECTOR-valued function. (TWO params since a 2D surface.) (2) dAreaScalar means: the area of the tiny parallelogram drawn on the surface, you get by varying p by dp and q by dq. dRvector dRvector dAreaScalar = | -------- X -------- | dp dq dp dq A FLUX-THRU-A-SURFACE integral (which is a double integral since a surface is TWO dimensional) is an integral / / | | F(x,y,z) . dAreaScalar = Flux of the vector-field F out of/thru the surface / / over a surface where F is a VECTOR-valued function where: (1) the surface has to be parameterized, i.e. (x,y,z) = R(p,q) where p and q are the two parameters and R is a VECTOR-valued function. (TWO params since a 2D surface.) (2) dAreaVector means: a vector, outward normal to the surface, whose length is dAreaScalar. dRvector dRvector dAreaVector = + -------- X -------- dp dq - dp dq (choose the +- sign to make it point outward.) A FLUX-THRU-A-CURVE INTEGRAL (which is a single integral since a curve is ONE dimensional) is the 2D version of the 3D flux--thru-a-surface-integral. Namely / | F(x,y,z) . dNormalLengthVector = Flux of the vector-field F out of/thru the curve / over a curve where F is a 2D-VECTOR-valued function where: (1) The curve has to be parameterized, i.e. (x,y,z) = R(t) where t is the parameter and F is a VECTOR-valued function. (2) dNormalLengthVector = (unitnormal vector pointing out) * dLengthScalar where dLengthScalar = |velocity| dt. This is a 90-degree rotated version of dLengthvector (tangential). You can rotate a vector (a,b) by 90 degrees by using (-b,a) or (b,-a) (choose which to make outward pointing). A CURVE INTEGRAL (OF A VECTOR) (which is a single integral since a curve is ONE dimensional) is an integral / | F(x,y,z) . dLengthVector = "Total push" of the vector-field F along the curve / over a curve where F is a VECTOR-valued function where (1) The curve has to be parameterized, i.e. (x,y,z) = R(t) where t is the parameter and F is a VECTOR-valued function. dRvector (2) dsVector = dLengthvector = -------- dt = VelocityVector dt (tangential) dt If this is a closed curve, this is often called a "circulation integral". A CURVE INTEGRAL (OF A SCALAR) (which is a single integral since a curve is ONE dimensional) is an integral / | F(x,y,z) . dLengthScalar / over a curve where F is a SCALAR-valued function where (1) The curve has to be parameterized, i.e. (x,y,z) = R(t) where t is the parameter and F is a VECTOR-valued function. (2) ds = dLengthScalar = |VelocityVector| dt = VelocityScalar dt. Important facts / labor saving tricks about these kinds of integrals: (1) If c is a CONSTANT then TRIPLE INTEGRAL of c dxdydz = c * THE VOLUME OF THE REGION DOUBLE INTEGRAL of c dxdy = c * THE AREA OF THE REGION SURFACE INTEGRAL of c dArea = c * THE AREA OF THE SURFACE CURVE INTEGRAL of c dArea = c * THE LENGTH OF THE CURVE If c is nonconstant (such as c=x) those won't work! (2) You can often use some kind of symmetry argument to save work. For example, to instantly realize some integral is 0 because of odd symmetry. Or for example, realize we have a spherically symmetric region and a spherically symmetric integrand, to reduce a triple integral to a single integral (over r). Or realize the flux is always normal to the surface, so that Fvec . dAreaVec = |F| dArea. Or realize the flux is always tangential to the surface, so Fvec . dAreaVec = 0. (3) NEWTON FORCE-ENERGY THEOREM (curve integrals of gradients): integral (grad H) . dTangentialLengthVector = H(B) - H(A). ANY curve segment from point A to point B (4) STOKES theorem relates circulation integrals round closed curves, to flux integrals thru (toplogically-disk) surfaces bounded by that curve. (Such as the northern hemisphere, bounded by the equatorial circle.) / / / | | (curl Fvec) . dAreaVector = | Fvec . dTangentialLengthVector = scalar / / / surface curve Signs: The "outward" normal direction of the surface should be selected so it is the direction your right thumb points, as your curled fingers point in the direction we go round the curve in the curve integral. Sometimes the surface integral is easier (select a good choice of surface - since all surfaces work, choose an easy one to handle) sometimes the curve integral is easier. (5) GREEN theorem is the 2D case of Stokes theorem, where the surface is flat in the XY plane and the curve is its boundary and F = (P,Q,0). (Such as the unit circle bounding the unit disk.) Things simplify since various things are 0, to: / / dP dQ / | | (-- - --) . dArea = | (P,Q) . dTangentialLengthVector = scalar / / dx dy / area curve (anticlockwise). (6) GAUSS DIVERGENCE THEOREM relates flux integral of vector field F out of a closed surface, to the volume integral of div F in the volume that surface encloses. (E.g. the surface of a sphere, and the region inside). / / / / / | | | (div Fvec) dx dy dz = | | Fvec . dAreaVector = scalar = flux of Fvec out of the surface / / / / / (outward normal) enclosed bounding volume surface (7) GAUSS DIVERGENCE THEOREM 2D VERSION is / / / | | (div Fvec) dx dy dz = | Fvec . dLengthVector = scalar = flux of Fvec out of the curve / / / (outward normal) enclosed bounding area curve Note, all the 3-7 involve 1 extra integral of something which has 1 extra derivative, a fact which is a useful mnemonic.