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微積分及應用
因舊版課程無指定課堂作業與考試,因此統整所有作業、講義、考試內容合併列出。
微積分及應用
因舊版課程無指定課堂作業與考試,因此統整所有作業、講義、考試內容合併列出。
使用須知:本部分的內容均以教科書的形式給出。課程分為不同章節,每章節包括很多網頁,使用網頁上的導航按紐可以進入下一章或回到上一層網頁。
18.013A 線上教科書Online Textbook (HTML®)
作業
複習習題1 - 5
習題1
給定以下五個向量:A = (1, 2, 3); B = (2, -3, 5); C = (x, y, z); D = (cos t, sin t, t 2); E = (-2, 1, 0).
做以下題目:
寫出A + B + C之總合。
計算 A•B.
計算A×(B+C).
求滿足C•A = 0和 C•B = 0時的 x,,y ,z的值。
求A 和B. B 和 D 所成角的餘弦值( 用t的函數表示)。
求 E 在 B上的投影。
求列式為A, B 和 E的行列式;再求出列式為 A, B 和 C的行列式。
假定點P的座標為x = 1, y = 2, z = 3,它的球面座標 ρ, θ 和 Φ為多少?
邊長為A,B,E的平形六面體的體積為多少?
求 D 在平面xy 上的投影。它的長度為多少?
習題2
已知直線上有點A和B
寫出該直線上兩點的參數表示形式。
寫出該直線上兩點處一單位長度的“法向量”。
求與該向量垂直的兩個方向。
已知平面上有點 A, B 和E:
寫出該平面的參數表示形式(兩個參數)。
求該平面的垂線。
求滿足該平面所有點的方程。
假定有一種新的不同與以往的向量內積V@W ,它具有以下性質:對於所有的V 和 @都有V@V = 0,且它們為線性變化,因此可以使用分配律
藉由(V + W)@(V + W)推導出與V@W + W@V相關的內容。
習題3
區分以下指定變數的函數:
sin (2x)
(sin xy)ex+y固定 y,關於x微分
x2 + y2 - 3xy固定 y,.關於x微分
(sin (y + s sin t))e-(x+s cos t).關於s微分,其他變數固定
求(sin y)e-x的梯度
求該函數在單位向量為 (cos t, sin t)的方向上的方向導數
sin (ex) 對於x=0的線性逼近
(r×v)對於t求導數, v 為dr/dt;假定 dv/dt 和 r的方向一致,那麼答案為多少?
1/x 為何不可微? tan x 為何不可微? |x| 為何不可微?
求sin (ex)反函數的導數(完整地定義一個反函數需要給出它的值域;在此處可以忽略 )。
習題4
求函數r = (x2 + y2)1/2 和 ρ = (x2 + y2 + z2)1/2的梯度。
求1/ρ的梯度。
求 cos θ和θ的梯度。
求 (y, z, x)的旋度。
求ρ/ρ3 的散度(記住ρ = (x, y, z))。
求它的旋度。
習題5
求sin xy在x = 1, y = 2 (弧度)時的二次線性逼近。
該函數在何處具有臨界點(兩側偏倒數均為 0)。
至少找出一個鞍點。
計算出(a×b)•(a×b)交換點和內積後的結果,用同樣的方法寫出三元內積的形式, 得到一個與內積形式等價的交錯式。
以下哪個函數在x = 0出有意義? (1-cos x)/x2, x2/sinx, (sin x cos x)/x2?
使用工具或軟體
.需要Java® plug-in software來執行本部分的Java®檔案
預備積分Precalculus
函數的運算 Operations on Functions
直線的斜率Slope of a Line
單變數的微積分Single Variable Calculus
導數與切線Derivative and Tangent Line
常數、一次、二次、三次線性逼近 Constant, Linear, Quadratic and Cubic Approximations
牛頓的方法 Newton's Method
雙變數的拉格朗日乘數Lagrange Multipliers with Two Variables
向量與代數Vectors and Algebra
轉動的座標系Rotating Coordinates
行列式與向量的積 Determinant and Vector Products
利用矩陣計算向量乘法Multiplication of a Vector by a Matrix
三維中的線性逼近 Linear Transformations in Three Dimensions
三維線性幾何中的應用Application to 3D Linear Geometry
空間中的線Lines in Space
空間中的面Planes in Space
曲線 Curves
極座標圖Polar Plotter
二維中的曲線 Curves in Two Dimensions
三維中的曲線Curves in Three Dimensions
場和曲面Fields and Surfaces
等高線,梯度與方向函數Contour Lines, Gradients and Directional Derivatives
曲線與曲面Curves and Surfaces
雙變數方程Functions of Two Variables
雙變數雙方程的牛頓的方法Newton's Method with Two Equations and Two Variables
複數與複數函數Complex Numbers and Functions
曲線的積分Integration on Curves
曲線與向量場Curves and Vector Fields
線積分 Line Integrals
面的積分Integration on Surfaces
流量積分Flux Integrals
積分的界限Integration Bounds
微分方程 Differential Equations
一階微分方程First Order ODE
二階微分方程Second Order ODE
一階微分方程組 System of First Order ODE
應用 Applications
二維中的靜電場Static Electric Fields in Two Dimensions
三維中的靜電場 Static Electric Fields in Three Dimensions
二維中的平穩磁場 Stationary Magnetic Fields in Two Dimensions
RLC(電阻-電感-電容)串聯電路 Series RLC Circuit
研習資料
符號辭彙表 |
|
符號 |
含義 |
i |
-1的平方根The square root of minus one |
f(x) |
自變數為x的函數f的值The value of the function f at argument x |
sin(x) |
自變數為x的正弦函數值 The value of the sine function at argument x |
exp(x) |
自變數為x的指數函數值,也記作ex The value of the exponential function at argument x. This is often written as ex |
a^x |
a的x次方;有理數x定義為反函數 The number a raised to the power x; for rational x is defined by inverse functions |
ln x |
exp x的反函數The inverse function to exp x |
ax |
與a^x相同Same as a^x |
logba |
為了得出a ,b; blogba = a The power you must raise b to in order to get a; blogba = a |
cos x |
自變數為x的餘弦函數的值(正弦的餘) The value of the cosine function (complement of the sine) at argument x |
tan x |
由 sin x/cos x得出Works out to be sin x/cos x |
cot x |
正切函數的餘的值或 cos x/sin x The value of the complement of the tangent function or cos x/sin x |
sec x |
正割函數的值,即 1/cos x Value of the secant function, which turns out to be 1/cos x |
csc x |
正割函數餘的值,即 1/sin x Value of the complement of the secant, called the cosecant. It is 1/sin x |
asin x |
自變數為x的正弦的反函數的值. 即 x = sin y The value, y, of the inverse function to the sine at argument x. Means x = sin y |
acos x |
自變數為x的餘弦的反函數的值. 即 x = cos y The value, y, of the inverse function to cosine at argument x. Means x = cos y |
atan x |
自變數為x的正切的反函數的值. 即 x = tan y The value, y, of the inverse function to tangent at argument x. Means x = tan y |
acot x |
自變數為x的餘切反函數的值. 即 x = cot y The value, y, of the inverse function to cotangent at argument x. Means x = cot y |
asec x |
自變數為x的正割的反函數的值. 即 x = sec y The value, y, of the inverse function to secant at argument x. Means x = sec y |
acsc x |
自變數為x的反割的反函數的值. 即 x = csc y The value, y, of the inverse function to cosecant at argument x. Means x = csc y |
θ |
角的標準符號. 除注明外一般用弧度制,特別用於atan x/y 當 x, y, z 是描述三維空間裏的點的時候 A standard symbol for angle. Measured in radians unless stated otherwise. Used especially for atan x/y when x, y, and z are variables used to describe point in three dimensional space |
i, j, k |
分別是x 、y、 z方向上的單位向量 Unit vectors in the x y and z directions respectively |
(a, b, c) |
x 分量為a,y分量為 b,z分量為c的向量 A vector with x component a, y component b and z component c |
(a, b) |
x 分量為a,y分量為 b的向量 A vector with x component a, y component b |
(a, b) |
向量a和b的內積The dot product of vectors a and b |
a•b |
向量a和b的內積The dot product of vectors a and b |
(a•b) |
向量a和b的內積The dot product of vectors a and b |
|v| |
向量v的量The magnitude of the vector v |
|x| |
X的絕對值The absolute value of the number x |
Σ |
表示和,它的參數和起始標號常標示在此符號的下方而終止標號則標示於其上方。 例如從j=1到n的和可寫為 , 意為1 + 2 + … + n Used to denote a summation, usually the index and often their end values are written under it with upper end value above it. For example the sum of j for j=1 to n is written as . This signifies 1 + 2 + … + n |
M |
用於表示數值或其他物件的矩陣或陣列。 Used to represent a matrix or array of numbers or other entities |
|v> |
行向量,其物件排列成行,可視為k乘1的矩陣。 A column vector, that is one whose components are written as a column and treated as a k by 1 matrix |
<v| |
列向量,或1乘k的矩陣。 A vector written as a row, or 1 by k matrix |
dx |
變數x的無窮小的變化; dy, dz, dr 等同理 An "infinitesimal" or very small change in the variable x; also similarly dy, dz, dr etc... |
ds |
非常小的距離變化A small change in distance |
ρ |
變數(x2 + y2 + z2)1/2 或球體座標系中到原點的距離 The variable (x2 + y2 + z2)1/2 or distance to the origin in spherical coordinates |
r |
變數(x2 + y2)1/2或在三維座標或極座標中到Z軸的距離 The variable (x2 + y2)1/2 or distance to the z axis in three dimensions or in polar coordinates |
|M| |
矩陣M的行列式( 由他的行或列確定面積或側面平行的區域的體積) The determinant of a matrix M (whose magnitude is the area or volume of the parallel sided region determined by its columns or rows) |
||M|| |
矩陣M的行列式的量,是體積,面積或超體積 The magnitude of the determinant of the matrix M, which is a volume or area or hypervolume |
det M |
M的行列式The determinant of M |
M-1 |
矩陣M的逆The inverse of the matrix M |
v×w |
向量v和向量w的內積或叉積。 The vector product or cross product of two vectors, v and w |
θvw |
向量v和w的夾角。The angle made by vectors v and w |
A•B×C |
標量三重積,或是由列A,B,C所構成矩陣的行列式值。 The scalar triple product, the determinant of the matrix formed by columns A, B, C |
uw |
向量w方向上的單位向量,它的含義與w/|w|相同 A unit vector in the direction of the vector w; it means the same as w/|w| |
df |
函數f的無窮小變化,小至對於所有函數皆可利用線性近似表示。 A very small change in the function f, sufficiently small that the linear approximation to all relevant functions holds for such changes |
df/dx |
f對x的微分,是對於函數f的線性逼近。 The derivative of f with respect to x, which is the slope of the linear approximation to f |
f ' |
f對變數的微分,通常對x。 The derivative of f with respect to the relevant variable, usually x |
∂f/∂x |
函數f對x的偏微分,固定y和z。一般來說函數f對某一指定變數q的偏微分 所代表的是在其餘變數固定的情形之下,df對dq的比值。 由於對於哪個變數保持固定可能產生誤解,所以應該明確標注出來。 The partial derivative of f with respect to x, keeping y, and z fixed. In general a partial derivative of f with respect to a variable q is the ratio of df to dq when certain other variables are held fixed. Where there is possible misunderstanding over which variables are to be fixed that information should be made explicit |
(∂f/∂x)|r,z |
固定r和z,函數f對x的偏微分。 The partial derivative of f with respect to x keeping r and z fixed |
grad f |
各分量分別為f對x,y和z的偏微分的向量場: [(∂f/∂x), (∂f/∂y), (∂f/∂z)] or (∂f/∂x)i + (∂f/∂y)j + (∂f/∂z)k;稱作f的梯度 The vector field whose components are the partial derivatives of the function f with respect to x, y and z: [(∂f/∂x), (∂f/∂y), (∂f/∂z)] or (∂f/∂x)i + (∂f/∂y)j + (∂f/∂z)k; called the gradient of f |
∇ |
向量運算(∂/∂x)i + (∂/∂x)j + (∂/∂x)k, 讀作 "del" The vector operator (∂/∂x)i + (∂/∂x)j + (∂/∂x)k, called "del" |
∇f |
f的梯度;它與uw的內積為f在w方向上的方向導數 The gradient of f; its dot product with uw is the directional derivative of f in the direction of w |
∇•w |
向量場w的散度;為向量算符∇ 與向量w的內積, 或(∂wx /∂x) + (∂wy /∂y) + (∂wz /∂z) The divergence of the vector field w; it is the dot product of the vector operator ∇ with the vector w, or (∂wx /∂x) + (∂wy /∂y) + (∂wz /∂z) |
curl w |
向量算符∇與向量w的叉積。 The cross product of the vector operator ∇ with the vector w |
∇×w |
w的旋度,分量為[(∂fz /∂y) - (∂fy /∂z), (∂fx /∂z) - (∂fz /∂x), (∂fy /∂x) - (∂fx /∂y)] The curl of w, with components [(∂fz /∂y) - (∂fy /∂z), (∂fx /∂z) - (∂fz /∂x), (∂fy /∂x) - (∂fx /∂y)] |
∇•∇ |
拉普拉斯運算元,為微分運算元:(∂2/∂x2) + (∂/∂y2) + (∂/∂z2) The Laplacian, the differential operator: (∂2/∂x2) + (∂/∂y2) + (∂/∂z2) |
f "(x) |
f對x的二階導數;亦為f '(x)的導數 The second derivative of f with respect to x; the derivative of f '(x) |
d2f/dx2 |
f對x的二階導數The second derivative of f with respect to x |
f(2)(x) |
f對x的二階導數的另一種形式 Still another form for the second derivative of f with respect to x |
f(k)(x) |
f對x的k階導數;亦為f(k-1) (x)的導數 The k-th derivative of f with respect to x; the derivative of f(k-1) (x) |
T |
沿著曲線的單位切向量;若曲線為r(t), T = (dr/dt)/|dr/dt| Unit tangent vector along a curve; if curve is described by r(t), T = (dr/dt)/|dr/dt| |
ds |
沿著曲線的長度的微分A differential of distance along a curve |
κ |
曲線的曲率;為單位切向量對距離取導數所得向量的大小: |dT/ds| The curvature of a curve; the magnitude of the derivative of its unit tangent vector with respect to distance on the curve: |dT/ds| |
N |
沿著dT/ds方向上的單位向量,與T垂直。 A unit vector in the direction of the projection of dT/ds normal to T |
B |
垂直與平面T 和 N的單位向量, 即面的曲率 A unit vector normal to the plane of T and N, which is the plane of curvature |
τ |
曲線的擾率; |dB/ds| The torsion of a curve; |dB/ds| |
g |
引力常數The gravitational constant |
F |
力學中力的標準符號The standard symbol for force in mechanics |
k |
彈簧的彈性係數The spring constant of a spring |
pi |
第I個質點的動量The momentum of the i-th particle |
H |
物理系統的哈密頓量,它的能量用{ri} 和 {pi}即 位置和動量表示 The Hamiltonian of a physical system, which is its energy expressed in terms of {ri} and {pi}, position and momentum |
{Q, H} |
Q 和 H的泊松括弧The Poisson bracket of Q and H |
|
f(x)的反導數, 用x的函數表示出來An antiderivative of f(x) expressed as a function of x |
|
f從a到b的定積分。當 f 為正值且 a < b , 那它代表 直線 y = a, y = b與x軸和函數曲線所圍成的面積 The definite integral of f from a to b. When f is positive and a < b holds, then this is the area between the x axis the lines y = a, y = b and the curve that represents the function f between these lines |
L(d) |
在等區間大小d下,f在每一區間中取左端點所得的黎曼和。 A Riemann sum with uniform interval size d and f evaluated at the left end of each subinterval |
R(d) |
在等區間大小d下,f在每一區間中取右端點所得的黎曼和。 A Riemann sum with uniform interval size d and f evaluated at the right end of each subinterval |
M(d) |
在等區間大小d下,f在每一區間中取極大值點所對應的點所得的黎曼和。 A Riemann sum with uniform interval size d and f evaluated at the maximum point of f in each subinterval |
m(d) |
在等區間大小d下,f在每一區間中取極小值點所對應的點所得的黎曼和。 A Riemann sum with uniform interval size d and f evaluated at the minimum point of f in each subinterval |