Back to main menu Summary of Math 223 Mathematical objects we have met 1. Scalar variables: These are variables which hold a single value, such as x, y, or t. 2. Vector variables (Chapter 12): These are variables which store several pieces of information and are denoted by $\vec{v} = v_1\hat{i} + v_2\hat{j} + v_3\hat{k}$ (in three space) or $\vec{v} = (v_1, v_2, \dots, v_n)$ (in n dimensional space.) 3. Scalar fields (Chapter 11): These are also referred to as functions. An object such as f(x,y,z) assigns a single scalar value to each point (x,y,z) in the domain of f. 4. Vector fields (17.1): A vector field such as $\vec{F}(x,y,z) = F_1(x,y,z)\hat{i} + F_2(x,y,z)\hat{j} + F_3(x,y,z)\hat{k}$ assigns a vector to each point in the domain. 5. Parameterized curves (16.1): A curve is a one dimensional object and can be described by a single parameter, such as t. To describe a curve, simply specify x = x(t), y = y(t), z = z(t) together with a range of values for t (such as $a \le t \le b$.) Optionally, one can use the position vector, $\vec{r}(t) = x(t)\hat{i} + y(t)\hat{j} + z(t)\hat{k}$. 6. Parameterized surfaces: see section 16.3 7. The del operator (13.4-5, 20.1, 20.3): (also called nabla) is the operator $\nabla = \frac{\partial}{\partial x}\hat{i} + \frac{\partial}{\partial y}\hat{j} + \frac{\partial}{\partial z}\hat{k}$. It has no meaning by itself and must be used to operate on other objects. Derivative operations on these objects Scalar fields: For a scalar field f(x,y,z) we have 1. Partial derviatives (13.1-2, 13.5): $f_x = \frac{\partial f}{\partial x}$denotes the rate of change of f in the x-direction, similar derivatives exist for the y and z directions; 2. Higher partial derivatives (13.7): since the partial derivatives are functions, they too have derivatives; if the function is smooth, then mixed partials are equal (ie. fxy = fyx, fxzz = fzxz = fzzx;) 3. Gradients (13.4-5): grad $f = \nabla f = f_x \hat{i} + f_y\hat{j} + f_z\hat{k}$. At each point, grad f tells you which direction f is increasing the most, as well as the rate of increase in that direction; 4. Directional derivative (13.4-5): the rate of change of a function f in the direction of the unit vector $\hat{u}$ is $f_{\hat{u}} = (\nabla f) \cdot \hat{u}$; 5. Laplacian (??): the Laplacian of a scalar field is simply $\nabla \cdot \nabla f = f_{xx} + f_{yy} + f_{zz}$. Vector fields 1. Divergence (20.1): The divergence of a vector field measures the flux density at a point. It is represented by div $\vec{F} = \nabla \cdot \vec{F}$ which is defined to be $$\nabla \cdot \vec{F}(x,y,z) = \lim_{\mbox{volume} \rightarro... ...{\int_S \vec{F} \cdot \hat{n}dS}{\mbox{volume enclosed by } S}.$$ 2. Curl (20.3): The curl of a vector field is a vector that points in the direction perpendicular to the plan of maximum circulation density and has magnitude equal to this maximum circulation density. In other words, it points in the direction in which $$\mbox{circ}_{\hat{n}} = \lim_{\mbox{area} \rightarrow 0} \frac{\int_C \vec{F} \cdot d\vec{r}}{\mbox{area enclosed by } C}$$ is the greatest. Parameterized curves: For a curve $\vec{r}(t)$ 1. Velocity (16.2): The velocity vector, $\vec{v}(t)$, points tangent to the path, and is simply $\frac{d\vec{r}}{dt} = x'(t)\hat{i} + y'(t)\hat{j} + z'(t)\hat{k}$. 2. Speed (16.2): This is the magnitude of the velocity vector, $s(t) = \vert\vert\vec{v}(t)\vert\vert$. 3. Acceleration (16.2): this measures the change in the velocity vector; $\vec{a}(t) = \frac{d\vec{v}}{dt}$. 4. Distance along the curve (16.2): if the path never crosses itself and there is no value of t for which $\vec{v}(t) = \vec{0}$ then the distance along the curve from t = a to t = b is $\int_a^b \vert\vert\vec{v}(t)\vert\vert dt$. Integral operations on these objects Scalar fields 1. The volume beneath the surface f(x,y) that lies above the region R (15.1) in the xy-plane is $\int_R f(x,y) dA$ where dA is the area element in the coordinate system. In Cartesian, dA = dx dy. In polar (15.5), $dA = rdr d\theta$. This is usually expressed as an iterated integral (15.2), for example: $$\int_R f dA = \int_{\mbox{bottom}}^{\mbox{top}} \int_{\mbox{left}}^{\mbox{right}} f(x,y) dx dy.$$ 2. Triple integrals (15.3): these are simply a further extension of double integrals. For an interpretion, let W be a three dimensional volume over which the mass density of an object, $\delta (x,y,z)$ is defined. Then $$\mbox{total mass} = \int_W \delta(x,y,z) dV$$ where dV is the appropriate volume element. In Cartesian, dV = dx dy dz. In cylindrical (15.6), $dV = r dr d\theta dz$. In spherical (15.6), $dV = \rho^2 \sin \phi d\rho d\phi d\theta$. 3. Averages: the average value of f(x,y) over the region R is $$\bar{f} = \frac{\int_R f dA}{\mbox{area of $R$}}.$$ If the function is defined over a three dimensional region W then the average of f over this region is $$\bar{f} = \frac{\int_W f dV}{\mbox{volume of $W$}}.$$ 4. Area of a region: the total area of R is simply $\int_R dA$. 5. Volume: the total volume of a region W is $\int_W dV$. Vector fields 1. Line integrals (18.1-2): These measure the amount of work which a vector field ``does'' along a curve. If the curve, C, is parameterized by $\vec{r}(t) = x(t)\hat{i} + y(t)\hat{j} + z(t)\hat{k}, a \le t \le b$ then $$\int_C \vec{F} \cdot d\vec{r} = \int_a^b \vec{F}(x(t),y(t),z(t)) \cdot \vec{v}(t) dt.$$ 2. Flux integrals (19.1-2): These measure how much a vector field flows through an oriented surface. If the surface, S, is given by z = f(x,y) and lies above a region R (this is the shadow of S) in the xy-plane then $$\int_S \vec{F} \cdot \hat{n} dS = \int_R \vec{F}(x,y,f(x,y)) \cdot (-f_x \hat{i} - f_y \hat{j} + \hat{k}) dA$$ if the surface is oriented upwards (towards +z axis.) Optimization Location of critical points (14.1) A point P is said to be a critical point of the function f(x,y,z) if either 1. $\nabla f (P) = \vec{0}$ (ie. fx(P) = fy(P) = fz(P) = 0), or 2. $\nabla f$ fails to exist at point P. Classification of critical points (14.2) Critical points come in three types. Maxima, minima, and saddle points. 1. A point P is a local maximum of f if f(x,y,z) < f(P) for all (x,y,z) near P. 2. A point P is a local minimum of f if f(x,y,z) > f(P) for all (x,y,z) near P. 3. A point P is a saddle point of f if there exist points Q and Q' near P such that f(P) > f(Q) and f(P) < f(Q'). The second derivative test can be applied to classify these points. Look at the quantities D = fxx fyy - fxy2 and fxx at the critical point P. If 1. D > 0, fxx < 0 then P is a local maximum, 2. D > 0, fxx > 0 then P is a local minimum, 3. D = 0 then P is a saddle point, 4. D < 0 then the second derivative test provides no information. Constrained optimization (14.3) Here we seek to find the maximum or minimum of some function f(x,y,z) subject to the constraint that g(x,y,z) = c. This will occur at solutions $(x,y,z; \lambda)$ of the Lagrange multiplier problem $$\nabla f(x,y,z) = \lambda \nabla g(x,y,z), \qquad g(x,y,z) = c.$$ In this case, if the point $(x(c),y(c),z(c); \lambda)$ is the solution for a given value of c then $$\lambda = \frac{d}{dc} f(x(c),y(c),z(c)).$$ In other words, it tells you the rate of change of the maximal value of f as the constraint increases (``how much bang for your buck.'') Special vector fields Conservative and gradient fields (18.3) A vector field $\vec{F}$ is conservative if any of the following conditions are met: 1. $\int_C \vec{F} \cdot d\vec{r}$ is independent of path, 2. the circulation of $\vec{F}$ around every closed path is zero, 3. $\vec{F} = \nabla f$ for some scalar potential function f(x,y,z). Further, the curl test (20.3,20.5)can be applied to certain vector fields to determine whether the vector field is conservative. If $\vec{F}$ is a smooth vector field in three-space so that 1. $\nabla \times \vec{F} = \vec{0}$, and 2. the domain,D, of $\vec{F}$ is such that any closed curve that lies in D can be smoothly contracted to a point while always remaining inside D, then $\vec{F}$ is a gradient field. Curl fields (20.3, 20.5) A vector field $\vec{F}$ is said to be a curl field if $\vec{F} = \nabla \times \vec{G}$ for some vector potential $\vec{G}$. The divergence test can help determine if $\vec{F}$ is a curl field. If $\vec{F}$ is a smooth vector field in three-space and 1. $\nabla \cdot \vec{F} = 0$, and 2. the domain, D, of $\vec{F}$ is such that every closed surface in D encloses only points that also lie in D, then $\vec{F}$ is a curl field. Integral theorems Fundamental Theorem of Calculus for Line Integrals (18.4) If C is a piecewise smooth oriented path from P to Q and f(x,y,z) is a function whose gradient is continuous on C then $$\int_C \nabla f \cdot d\vec{r} = f(Q) - f(P).$$ Divergence Theorem (20.2) If W is a solid region whose boundary S is piecewise smooth and oriented outward and $\vec{F}$ is a smooth vector field defined over all of S and W then $$\int_S \vec{F} \cdot \hat{n}dS = \int_W (\nabla \cdot \vec{F}) dV.$$ Green's Theorem (18.4) If C is a simple closed curve surrounding a region R in the plane, oriented so that R is always on the left as you traverse C and $\vec{F} = F_1 \hat{i} + F_2 \hat{j}$ is smooth and defined over all of R and C then $$\int_C \vec{F} \cdot d\vec{r} = \int_R \left( \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right) dA.$$ Stokes' Theorem (20.4) If S is a smooth oriented surface with a piecewise smooth boundary C oriented (from the orientation of S) by the right-hand rule and $\vec{F}$is a smooth vector field defined over all of S and C then $$\int_C \vec{F} \cdot d\vec{r} = \int_S (\nabla \times \vec{F}) \cdot \hat{n} dS.$$ Back to main menu