Introduction to vector calculus homework

By: Sheldon Austin


Introduction to vector calculus homework:

A Vector is a quantity of having magnitude and direction such as displacement, velocity, force, and acceleration. Let R (u) be a vector depending on a single scalar variable u, Then (ΔR)/(Δu) = (1/Δu) R (u + Δu) - R (u) where Δu denotes an increment in u. In integral of vectors, Let R(u) = R1(u)i + R2(u)j + R3(u)k be a vector depending on a scalar variable u, where R1(u), R2(u), R3(u) are supposed continuous in a specified interval. Then ∫ R (u) du = i ∫ R1 (u) du + j ∫ R2 (u) du +k ∫ R3 (u) du is called an indefinite integral of R (u).

vector calculus homework- formulas:

Differentiation formula:

1. ` d/(du)` (A + B) = `(dA)/(du)` + `(dB)/(du)`

2. ` d/(du)` (A . B) = A.( `(dB)/(du)`) + ( `(dA)/(du)`).B

3. ` d/(du)` (A × B) = A × (`(dB)/(du)`) + (`(dA)/(du)`) × B

4. ` d/(du)` (`phi` A) = `phi` (`(dA)/(du)`) + `(dphi)/(du)` A

5. ` d/(du)` (A . B × C) = A . B ×` (dC)/(du)` + A . (`(dB)/(du)`)× C + (`(dA)/(du)`) . B × C

6.` d/(du)` (A × B × C) = A × (B × (` (dC)/(du)` )) + A × ((`(dB)/(du)`) × C) + (`(dA)/(du)`) × (B × C)

Integration formula:

1. Line integrals: `oint` A.dx = `oint` A1 dx + A2 dy + A3 dz

2. Surface integrals: `int` `int` A. ds = `int` `int` A.n ds

3. Volume integrals: `int` `int` `int` A.dv and `int` `int` `int` Φ dv

Examples- vector calculus homework:

Vector calculus homework problem 1:

If A = (2x2y - x4) i + (exy - y sin x) j + (x2 cos y) k,

Find: a) `(dA)/(dx)` and `(d^2A)/(dx^2)`

b) `(dA)/(dy)` and `(d^2A)/(dy^2)`

Solution:

a) `(dA)/(dx)` = `d/dx` (2x2y - x4) i + `d/dx` (exy - y sin x) j + `d/dx` (x2 cos y) k,

= (4xy - 4x3) i + (yexy - y cos x) j + 2x cos y k

`(d^2A)/(dx^2)` = `d/dx ` (4xy - 4x3) i + `d/dx` (yexy - y cos x) j + `d/dx` (2x cos y) k

=(4y - 12 x2) i + (y2exy + y sinx) j + 2 cos y k

b) `(dA)/(dy)` = `d/dy` (2x2y - x4) i + `d/dy` (exy - y sin x) j +` d/dy` (x2 cos y) k,

= 2x2 i + (xexy - sin x) j - x2 sin y k

`(d^2A)/(dy^2)`= `d/dy` (2x2 )i + `d/dy ` (xexy - sin x) j - `d/dy` (x2 sin y) k

= 0 + x2exy j - x2 cos y k

= x2exy j - x2 cos y k

Answers for given vector calculus home work problem :

a) `(dA)/(dx)` = (4xy - 4x3) i + (yexy - y cos x) j + 2x cos y k

`(d^2A)/(dx^2)`= (4y - 12 x2) i + (y2exy + y sinx) j + 2 cos y k

b) `(dA)/(dy)` = 2x2 i + (xexy - sin x) j - x2 sin y k

`(d^2A)/(dy^2)`= x2exy j - x2 cos y k

Vector calculus homework problem 2:

If R(u) = (u - u2) i + 2u3 j - 3 k, Find: `int_1^2R(u)du`

Solution:

`int_1^2R(u)du` =[[`(u^2/2)` - `(u^3/3)` ] i + `u^2/2` j - 3u k + c]21

= [[`(2^2/2)` - `(2^3/3)` ] i + `2^4/2` j -3(2) k + c] - [[`(1^2/2)` - `(1^3/3)` ] i + `1^4/2` j -3(1) k + c]

= [[`(4/2)` - `(8/3)` ] i + `16/2` j - 6 k + c] - [[`(1/2)` - `(1/3)` ] i + `1/2` j - 3k + c]

= [`(-2/3)` i + 8 j - 6 k + c] - [`(1/6)` i + 1/2 j - 3k + c]

= `(-5/6)` i + `(15/2)` j - 3k

Answers for given vector calculus home work problem : `(-5/6)` i + `(15/2)` j - 3k

Vector calculus home work practice problem:

Vector calculus home work practice problem 1:

Given R = sin t i + cos t j + t k

Find: `(d^2R)/dt^2`

Answer for given vector calculus home work practice problem 1:

-sin t i - cos t j

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