Calculus, Vector Calculus and Differential Equations

Limit evaluation

A limit describes the value that a function approaches as its input approaches a point. The function does not need to be defined at that point, and only the values nearby matter. If direct substitution gives a finite number, that number is the limit. If it gives \frac{0}{0}, the expression is an indeterminate form and must be rewritten first. Common methods are cancelling a common factor, rationalising, applying L'Hôpital's rule, or using a substitution that turns the expression into a standard limit. A cancellation is valid because it holds for every x \neq a, and the limit depends only on those nearby values. The standard limits \lim_{x\to 0}\frac{\sin x}{x}=1 and \lim_{x\to 0}\frac{\ln(1+x)}{x}=1 are used constantly in exams.

• indeterminate form — a form such as \frac{0}{0} that does not by itself fix the limit. Ex) \lim_{x\to 2}\frac{x^2-4}{x-2} becomes \lim_{x\to 2}(x+2)=4 after factoring.
• substitution to a standard limit — shifting the variable so the point moves to zero. Ex) For \lim_{x\to\pi/2}\frac{\cos x}{x-\pi/2}, put t=x-\pi/2, use \cos(\tfrac{\pi}{2}+t)=-\sin t, and the limit is -1.

For a vector-valued function \mathbf{r}(t), the limit is taken component by component, provided each component has a limit. A component such as \frac{\sin at}{t} tends to a, and a polynomial component tends to its value at the point.

When the leading terms cancel, look at the Taylor expansion. Since \sin u = u-\frac{u^3}{6}+\cdots, the difference of a function and its argument is of higher order than the denominator, so the limit can be zero even though both parts look large. Finally, if a question asks for a value correct to a given number of decimal places, find the exact limit first and round it at the end.

Extrema of functions

An extremum is a maximum or minimum value of a function. A critical point is a point where f'(x)=0 or f'(x) does not exist. A local extremum must be a critical point or an endpoint of the domain, but not every critical point is an extremum. On a closed interval [a,b], a continuous function attains its global maximum and minimum either at a critical point inside the interval or at an endpoint. The procedure is to list the critical points, evaluate f at each of them and at both endpoints, and compare the values.

The second derivative test classifies a critical point. If f'(c)=0 and f''(c)>0, then c is a local minimum. If f''(c)<0, then c is a local maximum. If f''(c)=0, the test is inconclusive and the sign change of f' must be examined. Example: for f(x)=x^3-3x, the critical points are x=\pm1, and f''(x)=6x gives f''(1)=6>0, so x=1 is a local minimum with value -2.

• local minimum — a value no smaller than the values at nearby points.
• global minimum on an interval — the smallest value among all critical values and endpoint values.

Homogeneous linear ODE with constant coefficients

A second-order homogeneous linear equation ay''+by'+cy=0 with constant coefficients is solved by trying y=e^{rx}. This gives the characteristic equation ar^2+br+c=0. If the roots are distinct and real, r_1 and r_2, the general solution is y=Ae^{r_1x}+Be^{r_2x}. If the root is repeated, the solution is (A+Bx)e^{rx}. If the roots are complex, \alpha\pm i\beta, the solution is e^{\alpha x}(A\cos\beta x+B\sin\beta x). The constants A and B are fixed by the initial conditions, which give two equations for two unknowns.

For an initial value problem with real distinct roots, the constants are obtained by substituting y(0) and y'(0) into the general solution and its derivative. Each candidate answer can be checked the same way: substitute x=0 into the function and into its derivative, and confirm both initial values. Example: for y''-y=0 with y(0)=1 and y'(0)=0, the roots are \pm1, the constants are A=B=\tfrac12, and y=\cosh x.

• characteristic equation — the algebraic equation ar^2+br+c=0 obtained by substituting y=e^{rx}.
• initial value problem — an ODE together with conditions on y and its derivative at the same point.

Directional derivative

The directional derivative of a scalar field f in the direction of a unit vector \hat{u} is D_{\hat{u}}f=\nabla f\cdot\hat{u}. The direction must be a unit vector, so a given vector \vec{V} must be divided by its magnitude first. The gradient is \nabla f=\left(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z}\right). When the direction is \hat{i}, the directional derivative is just the partial derivative \partial f/\partial x, and variables that do not change along that direction have no effect on the result.

Example: for f=xy at (1,2) along \vec{V}=3\hat{i}+4\hat{j}, the gradient is (2,1), the unit vector is (0.6,0.8), and the derivative is 1.2+0.8=2. Since the derivative is largest when \hat{u} points along \nabla f, the maximum rate of change equals |\nabla f|.

• unit vector — a vector of length 1, obtained by dividing a vector by its magnitude.
• gradient — the vector of first partial derivatives, pointing in the direction of steepest increase.

Curl of vector field

The curl of \vec{v}=P\hat{i}+Q\hat{j}+R\hat{k} is the determinant-form vector \nabla\times\vec{v}=\left(\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z}\right)\hat{i}+\left(\frac{\partial P}{\partial z}-\frac{\partial R}{\partial x}\right)\hat{j}+\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)\hat{k}. Each component is computed separately, and it is easy to reverse the sign of a cross term, so each component must be checked. A field whose components depend only on z has nonzero curl only through terms that involve \partial/\partial z. Example: for \vec{v}=y\hat{i}-x\hat{j}, the \hat{k} component is -1-1=-2, so the curl is -2\hat{k}.

Stokes' theorem states that the circulation of \vec{F} around a closed boundary equals the flux of \nabla\times\vec{F} through any surface bounded by that curve, \oint_C\vec{F}\cdot d\vec{r}=\iint_S(\nabla\times\vec{F})\cdot\hat{n}\,dS. For a flat surface in the XY plane, \hat{n}=\hat{k}, so only the \hat{k} component of the curl matters.

• circulation — the line integral of a vector field around a closed curve.
• flux — the surface integral of the normal component of a vector field.

Divergence of vector field

The divergence of \vec{V}=V_x\hat{i}+V_y\hat{j}+V_z\hat{k} is the scalar \nabla\cdot\vec{V}=\frac{\partial V_x}{\partial x}+\frac{\partial V_y}{\partial y}+\frac{\partial V_z}{\partial z}. It measures the net outflow per unit volume at a point. To evaluate it at a point, first differentiate each component with respect to its own variable, then substitute the coordinates. Example: for \vec{V}=x^2\hat{i}+yz\hat{j}, the divergence is 2x+z, which equals 5 at (1,2,3).

The divergence operator is linear, so \nabla\cdot(\vec{U}+\vec{V})=\nabla\cdot\vec{U}+\nabla\cdot\vec{V}, and constants pass through it: \nabla\cdot(c\vec{U})=c\,\nabla\cdot\vec{U}. The divergence of a sum is the sum of the divergences, not the divergence of one term multiplied by the other. Operators such as the curl are not used in this expansion, so options that mix curl and divergence are wrong.

• linearity — the property L(a\vec{U}+b\vec{V})=aL(\vec{U})+bL(\vec{V}) for a linear operator L.
• outflow — the net flux of a field out of a small volume around a point.