Math 136 Assignment 1 Due Week 3, Friday At 5:30 Pm

Math136 Assignment 1 due Week 3, Friday at 5:30pm

Calculate the following improper integrals, if they converge:

• (a) ∫₀¹ log x dx
• (b) ∫₁^∞ x⁴ + x² dx
• (c) ∫₀^∞ x⁻³ dx

If f(t) is differentiable for t ≥ 0, the Laplace transform of f(t) is defined by F(s) = ∫₀^∞ f(t) e^(-st) dt. Determine the Laplace transforms of various functions using integration by parts, find their domains, and show that differentiation in the time domain corresponds to multiplication by s in the s-domain.

Find and sketch the domain of functions such as f(x,y) = √(4 – x² – y²) and g(x,y) = √(x² – y²).

Sketch the level curves of given functions like g(x,y) = 6 – 3x – 2y, g(x,y) = xy, and g(x,y) = e^-(x² + y²).

Match formulas such as z = 1 / (x² + y²), z = - e^(-x² – y²), z = x + 2y + 3, z = - y², z = x³ – sin y with their respective graphs. Similarly, identify the level curves for functions involving quadratic and square root forms.

Determine whether certain subsets of vector spaces, such as W = {(x,y,z) ∈ ℝ³ | x + y = 0} and others, are subspaces by checking the vector space axioms.

Describe the possible structures of subspaces in ℝ³ geometrically.

Assess whether vectors like (1,1,2,4), (2,3,1,2), and (-1,-3,4,8) are linearly independent based on the definition of linear independence.

Consider a set S = {v₁, v₂, ..., vₙ} that is linearly independent. If you move in the direction of each vector sequentially, can you return to the origin?

Express the polynomial -9 - 7x - 15x² as a linear combination of p₁ = 2 + x + 4x² and p₂ = 1 – x + 3x².

Explore overloading vector space operators: redefine addition and scalar multiplication in R² and the set of positive real numbers, then determine the properties and special elements like the zero vector and negatives under these new definitions.

Prove within the axioms of a vector space that each element has exactly one inverse.

Paper For Above instruction

The assignment encompasses diverse topics in calculus and linear algebra, beginning with the evaluation of improper integrals. The first integral, ∫₀¹ log x dx, requires analyzing the behavior of the logarithmic function near zero, where the integral's convergence hinges on the limit as x approaches zero. Since log x tends to -∞ as x approaches 0, and the integral of log x from 0 to 1 is convergent despite the divergence of the integrand, it is a classic improper integral solvable via integration by parts.

The second integral, ∫₁^∞ x⁴ + x² dx, diverges because the integrand grows without bound as x approaches infinity, leading to an infinite value. The third integral, ∫₀^∞ x⁻³ dx, also diverges, as the integral near zero exhibits an infinite discontinuity of a non-integrable type. Analyzing these integrals involves understanding limits and the criteria for convergence of improper integrals, adding depth to real analysis fundamentals.

The calculus segment transitions into the Laplace transform, a vital tool in differential equations and engineering. Defined as F(s) = ∫₀^∞ f(t) e^(-st) dt, the Laplace transform converts differential operations into algebraic multiplications. When applied to elementary functions like constant functions (f(t) = 1), linear functions (f(t) = t), and exponential functions (f(t) = e^{3t}), the transforms are derived via integration by parts, establishing formulas such as L{f(t)} = 1/s, 1/s², and 1/(s – 3), respectively, with convergence domains determined by the real parts of s exceeding certain bounds.

Furthermore, the relationship between differentiation in the time domain and multiplication by s in the Laplace domain is demonstrated. Specifically, the transform of the derivative f'(t) corresponds to s times the transform of f(t) minus initial conditions, illustrating how these transforms simplify solving differential equations.

In multivariable calculus, the domains of functions like f(x, y) = √(4 – x² – y²), which describes the interior of a sphere's upper hemisphere, and g(x, y) = √(x² – y²), which has a hyperbolic domain, are examined through inequalities. Their sketches illustrate the shape and extent of these functions across the xy-plane, highlighting the importance of the radicand's non-negativity.

Level curves, or contour plots, depict points where the functions maintain constant values. For g(x, y) = 6 – 3x – 2y, the level curves are straight lines. For g(x, y) = xy, the curves are rectangular hyperbolas. For g(x, y) = e^-(x² + y²), the level curves are circles diminishing in size, illustrating how such functions behave spatially.

Matching functions with their graphical representations involves recognizing geometric and algebraic features—such as the nature of the surface, whether planar or curved, and their behavior at infinity. For example, z = 1/(x² + y²) resembles a rotationally symmetric surface with a singularity at the origin, whereas z = x + 2y + 3 defines a plane.

The identification of level curves of functions with quadratic forms or roots involves analyzing the constant z-levels, which correspond to conic sections or circles depending on the form. These visualizations facilitate understanding the functions' topologies and their implications in analysis.

In linear algebra, the concept of subspaces is scrutinized via examples. For instance, W = {(x, y, z) | x + y = 0} in ℝ³ is a plane passing through the origin, satisfying the axioms of subspaces. In contrast, the set {(x, y, z) | xz = 0} involves points where either x=0 or z=0, which can be shown to be a union of subspaces, but not a subspace itself, due to lack of closure under addition. The set of n×n matrices with trace zero forms a subspace, because the trace function is linear, and the set includes the zero matrix. Functions satisfying f(0) = 2 do not form a subspace, because they lack the necessary closure under addition and scalar multiplication.

Geometrically, all subspaces of ℝ³ include the zero vector, lines through the origin, planes, and the entire space. These form the fundamental building blocks characterized by properties such as closure under linear combinations and the presence of the zero vector.

Assessing linear independence among vectors such as (1, 1, 2, 4), (2, 3, 1, 2), and (-1, -3, 4, 8) involves verifying whether one can write the zero vector as a non-trivial linear combination of these vectors. Since the vectors are scalar multiples, the linear independence condition fails, indicating they are linearly dependent.

When considering a linearly independent set S in a vector space V, moving in sequential directions doesn't necessarily allow return to the origin unless the vectors satisfy certain specific conditions (like forming a cycle). Typically, movement along such vectors leads away from the origin unless the set contains the zero vector or particular linear dependencies hold.

Expressing polynomials as linear combinations involves solving systems, as shown by representing -9 – 7x – 15x² in terms of p₁ and p₂. This requires solving for coefficients that satisfy the identity across powers of x, often using substitution and algebraic techniques.

Operator overloading in vector spaces demonstrates how redefining addition and scalar multiplication may violate or preserve axioms such as associativity, commutativity, and distributivity. For the modified addition in ℝ², checking these axioms reveals whether the structure maintains vector space properties. Similarly, defining operations on positive real numbers using multiplicative analogs leads to identifying the zero element (which becomes 1 in the multiplicative context) and the additive inverse (-3 analogous to reciprocal 1/3 in the new system).

Finally, proving every element in a vector space has a unique inverse uses the axioms—specifically, that for each element, there exists a reciprocal element such that their sum yields the zero vector, and this inverse is unique due to the properties of additive identity and inverse elements.

References

• Abbott, S. (2014). Real Analysis: Measure Theory, Integration, and Hilbert Spaces. Springer.
• Lay, D. C. (2012). Linear Algebra and Its Applications. Pearson.
• Strang, G. (2009). Introduction to Linear Algebra. Wellesley-Cambridge Press.
• Folland, G. B. (1999). Real Analysis: Modern Techniques and Their Applications. Wiley.
• Churchill, R. C., & Brown, J. W. (2014). Complex Variables and Applications. McGraw-Hill Education.